diff --git a/src/Lean/Elab/App.lean b/src/Lean/Elab/App.lean
index 7752ac82024f..3a9d32af64c9 100644
--- a/src/Lean/Elab/App.lean
+++ b/src/Lean/Elab/App.lean
@@ -1135,24 +1135,29 @@ private def throwLValError (e : Expr) (eType : Expr) (msg : MessageData) : TermE
   throwError "{msg}{indentExpr e}\nhas type{indentExpr eType}"
 
 /--
-`findMethod? env S fName`.
-- If `env` contains `S ++ fName`, return `(S, S++fName)`
-- Otherwise if `env` contains private name `prv` for `S ++ fName`, return `(S, prv)`, o
-- Otherwise for each parent structure `S'` of  `S`, we try `findMethod? env S' fname`
+`findMethod? S fName` tries the following for each namespace `S'` in the resolution order for `S`:
+- If `env` contains `S' ++ fName`, returns `(S', S' ++ fName)`
+- Otherwise if `env` contains private name `prv` for `S' ++ fName`, returns `(S', prv)`
 -/
-private partial def findMethod? (env : Environment) (structName fieldName : Name) : Option (Name × Name) :=
-  let fullName := structName ++ fieldName
-  match env.find? fullName with
-  | some _ => some (structName, fullName)
-  | none   =>
+private partial def findMethod? (structName fieldName : Name) : MetaM (Option (Name × Name)) := do
+  let env ← getEnv
+  let find? structName' : MetaM (Option (Name × Name)) := do
+    let fullName := structName' ++ fieldName
+    if env.contains fullName then
+      return some (structName', fullName)
     let fullNamePrv := mkPrivateName env fullName
-    match env.find? fullNamePrv with
-    | some _ => some (structName, fullNamePrv)
-    | none   =>
-      if isStructure env structName then
-        (getStructureSubobjects env structName).findSome? fun parentStructName => findMethod? env parentStructName fieldName
-      else
-        none
+    if env.contains fullNamePrv then
+      return some (structName', fullNamePrv)
+    return none
+  -- Optimization: the first element of the resolution order is `structName`,
+  -- so we can skip computing the resolution order in the common case
+  -- of the name resolving in the `structName` namespace.
+  find? structName <||> do
+    let resolutionOrder ← if isStructure env structName then getStructureResolutionOrder structName else pure #[structName]
+    for h : i in [1:resolutionOrder.size] do
+      if let some res ← find? resolutionOrder[i] then
+        return res
+    return none
 
 /--
   Return `some (structName', fullName)` if `structName ++ fieldName` is an alias for `fullName`, and
@@ -1204,7 +1209,7 @@ private def resolveLValAux (e : Expr) (eType : Expr) (lval : LVal) : TermElabM L
   | some structName, LVal.fieldName _ fieldName _ _ =>
     let env ← getEnv
     let searchEnv : Unit → TermElabM LValResolution := fun _ => do
-      if let some (baseStructName, fullName) := findMethod? env structName (.mkSimple fieldName) then
+      if let some (baseStructName, fullName) ← findMethod? structName (.mkSimple fieldName) then
         return LValResolution.const baseStructName structName fullName
       else if let some (structName', fullName) := findMethodAlias? env structName (.mkSimple fieldName) then
         return LValResolution.const structName' structName' fullName
@@ -1390,19 +1395,17 @@ private def elabAppLValsAux (namedArgs : Array NamedArg) (args : Array Arg) (exp
       loop f lvals
     | LValResolution.projFn baseStructName structName fieldName =>
       let f ← mkBaseProjections baseStructName structName f
-      if let some info := getFieldInfo? (← getEnv) baseStructName fieldName then
-        if isPrivateNameFromImportedModule (← getEnv) info.projFn then
-          throwError "field '{fieldName}' from structure '{structName}' is private"
-        let projFn ← mkConst info.projFn
-        let projFn ← addProjTermInfo lval.getRef projFn
-        if lvals.isEmpty then
-          let namedArgs ← addNamedArg namedArgs { name := `self, val := Arg.expr f, suppressDeps := true }
-          elabAppArgs projFn namedArgs args expectedType? explicit ellipsis
-        else
-          let f ← elabAppArgs projFn #[{ name := `self, val := Arg.expr f, suppressDeps := true }] #[] (expectedType? := none) (explicit := false) (ellipsis := false)
-          loop f lvals
+      let some info := getFieldInfo? (← getEnv) baseStructName fieldName | unreachable!
+      if isPrivateNameFromImportedModule (← getEnv) info.projFn then
+        throwError "field '{fieldName}' from structure '{structName}' is private"
+      let projFn ← mkConst info.projFn
+      let projFn ← addProjTermInfo lval.getRef projFn
+      if lvals.isEmpty then
+        let namedArgs ← addNamedArg namedArgs { name := `self, val := Arg.expr f, suppressDeps := true }
+        elabAppArgs projFn namedArgs args expectedType? explicit ellipsis
       else
-        unreachable!
+        let f ← elabAppArgs projFn #[{ name := `self, val := Arg.expr f, suppressDeps := true }] #[] (expectedType? := none) (explicit := false) (ellipsis := false)
+        loop f lvals
     | LValResolution.const baseStructName structName constName =>
       let f ← if baseStructName != structName then mkBaseProjections baseStructName structName f else pure f
       let projFn ← mkConst constName
diff --git a/src/Lean/Elab/Structure.lean b/src/Lean/Elab/Structure.lean
index 4a8db09ad52d..be5c1f8f0970 100644
--- a/src/Lean/Elab/Structure.lean
+++ b/src/Lean/Elab/Structure.lean
@@ -22,7 +22,12 @@ namespace Lean.Elab.Command
 
 register_builtin_option structureDiamondWarning : Bool := {
   defValue := false
-  descr    := "enable/disable warning messages for structure diamonds"
+  descr    := "if true, enable warnings when a structure has diamond inheritance"
+}
+
+register_builtin_option structure.strictResolutionOrder : Bool := {
+  defValue := false
+  descr := "if true, require a strict resolution order for structures"
 }
 
 open Meta
@@ -943,6 +948,23 @@ private def mkInductiveType (view : StructView) (indFVar : Expr) (levelNames : L
     instantiateMVars (← mkForallFVars params type)
   return { name := view.declName, type := ← instantiateMVars type, ctors := [{ ctor with type := ← instantiateMVars ctorType }] }
 
+/--
+Precomputes the structure's resolution order.
+Option `structure.strictResolutionOrder` controls whether to create a warning if the C3 algorithm failed.
+-/
+private def checkResolutionOrder (structName : Name) : TermElabM Unit := do
+  let resolutionOrderResult ← computeStructureResolutionOrder structName (relaxed := !structure.strictResolutionOrder.get (← getOptions))
+  trace[Elab.structure.resolutionOrder] "computed resolution order: {resolutionOrderResult.resolutionOrder}"
+  unless resolutionOrderResult.conflicts.isEmpty do
+    let mut defects : List MessageData := []
+    for conflict in resolutionOrderResult.conflicts do
+      let parentKind direct := if direct then "parent" else "indirect parent"
+      let conflicts := conflict.conflicts.map fun (isDirect, name) =>
+        m!"{parentKind isDirect} '{MessageData.ofConstName name}'"
+      defects := m!"- {parentKind conflict.isDirectParent} '{MessageData.ofConstName conflict.badParent}' \
+        must come after {MessageData.andList conflicts.toList}" :: defects
+    logWarning m!"failed to compute strict resolution order:\n{MessageData.joinSep defects.reverse "\n"}"
+
 def mkStructureDecl (vars : Array Expr) (view : StructView) : TermElabM Unit := Term.withoutSavingRecAppSyntax do
   let scopeLevelNames ← Term.getLevelNames
   let isUnsafe := view.modifiers.isUnsafe
@@ -1008,6 +1030,8 @@ def mkStructureDecl (vars : Array Expr) (view : StructView) : TermElabM Unit :=
             else
               mkCoercionToCopiedParent levelParams params view parent.structName parent.type
           setStructureParents view.declName parentInfos
+          checkResolutionOrder view.declName
+
           let lctx ← getLCtx
           /- The `lctx` and `defaultAuxDecls` are used to create the auxiliary "default value" declarations
             The parameters `params` for these definitions must be marked as implicit, and all others as explicit. -/
@@ -1045,6 +1069,8 @@ def elabStructure (modifiers : Modifiers) (stx : Syntax) : CommandElabM Unit :=
     pure view
   elabStructureViewPostprocessing view
 
-builtin_initialize registerTraceClass `Elab.structure
+builtin_initialize
+  registerTraceClass `Elab.structure
+  registerTraceClass `Elab.structure.resolutionOrder
 
 end Lean.Elab.Command
diff --git a/src/Lean/Server/Completion.lean b/src/Lean/Server/Completion.lean
index 83a38dac54aa..75623140c28c 100644
--- a/src/Lean/Server/Completion.lean
+++ b/src/Lean/Server/Completion.lean
@@ -535,7 +535,7 @@ where
     let .const typeName _ := type.getAppFn | return ()
     modify fun s => s.insert typeName
     if isStructure (← getEnv) typeName then
-      for parentName in getAllParentStructures (← getEnv) typeName do
+      for parentName in (← getAllParentStructures typeName) do
         modify fun s => s.insert parentName
     let some type ← unfoldeDefinitionGuarded? type | return ()
     visit type
diff --git a/src/Lean/Structure.lean b/src/Lean/Structure.lean
index 089d7f7c78fd..817da6c82be9 100644
--- a/src/Lean/Structure.lean
+++ b/src/Lean/Structure.lean
@@ -154,6 +154,7 @@ def getStructureCtor (env : Environment) (constName : Name) : ConstructorVal :=
 def getStructureFields (env : Environment) (structName : Name) : Array Name :=
   (getStructureInfo env structName).fieldNames
 
+/-- Get the `StructureFieldInfo` for the given direct field of the structure. -/
 def getFieldInfo? (env : Environment) (structName : Name) (fieldName : Name) : Option StructureFieldInfo :=
   if let some info := getStructureInfo? env structName then
     info.fieldInfo.binSearch { fieldName := fieldName, projFn := default, subobject? := none, binderInfo := default } StructureFieldInfo.lt
@@ -180,21 +181,7 @@ If a direct parent cannot itself be represented as a subobject,
 sometimes one of its parents (or one of their parents, etc.) can.
 -/
 def getStructureSubobjects (env : Environment) (structName : Name) : Array Name :=
-  let fieldNames := getStructureFields env structName;
-  fieldNames.foldl (init := #[]) fun acc fieldName =>
-      match isSubobjectField? env structName fieldName with
-      | some parentStructName => acc.push parentStructName
-      | none                  => acc
-
--- TODO: use actual parents, not just subobjects.
-/-- Return all parent structures -/
-partial def getAllParentStructures (env : Environment) (structName : Name) : Array Name :=
-  visit structName |>.run #[] |>.2
-where
-  visit (structName : Name) : StateT (Array Name) Id Unit := do
-    for p in getStructureSubobjects env structName do
-      modify fun s => s.push p
-      visit p
+  (getStructureFields env structName).filterMap (isSubobjectField? env structName)
 
 /--
 Return the name of the structure that contains the field relative to structure `structName`.
@@ -269,18 +256,23 @@ partial def getPathToBaseStructureAux (env : Environment) (baseStructName : Name
   if baseStructName == structName then
     some path.reverse
   else
-    let fieldNames := getStructureFields env structName;
-    fieldNames.findSome? fun fieldName =>
-      match isSubobjectField? env structName fieldName with
-      | none                  => none
-      | some parentStructName =>
-        match getProjFnForField? env structName fieldName with
-        | none        => none
-        | some projFn => getPathToBaseStructureAux env baseStructName parentStructName (projFn :: path)
+    if let some info := getStructureInfo? env structName then
+      -- Prefer subobject projections
+      (info.fieldInfo.findSome? fun field =>
+        match field.subobject? with
+        | none                  => none
+        | some parentStructName => getPathToBaseStructureAux env baseStructName parentStructName (field.projFn :: path))
+      -- Otherwise, consider other parents
+      <|> info.parentInfo.findSome? fun parent =>
+        if parent.subobject then
+          none
+        else
+          getPathToBaseStructureAux env baseStructName parent.structName (parent.projFn :: path)
+    else none
 
 /--
-If `baseStructName` is an ancestor structure for `structName`, then returns a sequence of projection functions
-to go from `structName` to `baseStructName`.
+If `baseStructName` is an ancestor structure for `structName`, then return a sequence of projection functions
+to go from `structName` to `baseStructName`. Returns `[]` if `baseStructName == structName`.
 -/
 def getPathToBaseStructure? (env : Environment) (baseStructName : Name) (structName : Name) : Option (List Name) :=
   getPathToBaseStructureAux env baseStructName structName []
@@ -315,4 +307,132 @@ def getStructureLikeNumFields (env : Environment) (constName : Name) : Nat :=
     | _ => 0
   | _ => 0
 
+/-!
+### Resolution orders
+
+This section is for computations to determine which namespaces to visit when resolving field notation.
+While the set of namespaces is clear (after a structure's namespace, it is the namespaces for *all* parents),
+the question is the order to visit them in.
+
+We use the C3 superclass linearization algorithm from Barrett et al., "A Monotonic Superclass Linearization for Dylan", OOPSLA 1996.
+For reference, the C3 linearization is known as the "method resolution order" (MRO) [in Python](https://docs.python.org/3/howto/mro.html).
+
+The basic idea is that we want to find a resolution order with the following property:
+For each structure `S` that appears in the resolution order, if its direct parents are `P₁ .. Pₙ`,
+then `S P₁ ... Pₙ` forms a subsequence of the resolution order.
+
+This has a stability property where if `S` extends `S'`, then the resolution order of `S` contains the resolution order of `S'` as a subsequence.
+It also has the key property that if `P` and `P'` are parents of `S`, then we visit `P` and `P'` before we visit the shared parents of `P` and `P'`.
+
+Finding such a resolution order might not be possible.
+Still, we can enable a relaxation of the algorithm by ignoring one or more parent resolution orders, starting from the end.
+
+In Hivert and Thiéry "Controlling the C3 super class linearization algorithm for large hierarchies of classes"
+https://arxiv.org/pdf/2401.12740 the authors discuss how in SageMath, which has thousands of classes,
+C3 can be difficult to control, since maintaining correct direct parent orders is a burden.
+They give suggestions that have worked for the SageMath project.
+We may consider introducing an environment extension with ordering hints to help guide the algorithm if we see similar difficulties.
+-/
+
+structure StructureResolutionState where
+  resolutions : PHashMap Name (Array Name) := {}
+  deriving Inhabited
+
+/--
+We use an environment extension to cache resolution orders.
+These are not expensive to compute, but worth caching, and we save olean storage space.
+-/
+builtin_initialize structureResolutionExt : EnvExtension StructureResolutionState ←
+  registerEnvExtension (pure {})
+
+/-- Gets the resolution order if it has already been cached. -/
+private def getStructureResolutionOrder? (env : Environment) (structName : Name) : Option (Array Name) :=
+  (structureResolutionExt.getState env).resolutions.find? structName
+
+/-- Caches a structure's resolution order. -/
+private def setStructureResolutionOrder [MonadEnv m] (structName : Name) (resolutionOrder : Array Name) : m Unit :=
+  modifyEnv fun env => structureResolutionExt.modifyState env fun s =>
+    { s with resolutions := s.resolutions.insert structName resolutionOrder }
+
+/-- "The `badParent` must come after the `conflicts`. -/
+structure StructureResolutionOrderConflict where
+  isDirectParent : Bool
+  badParent : Name
+  /-- Conflicts that must come before `badParent`. The flag is whether it is a direct parent. -/
+  conflicts : Array (Bool × Name)
+  deriving Inhabited
+
+structure StructureResolutionOrderResult where
+  resolutionOrder : Array Name
+  conflicts : Array StructureResolutionOrderConflict := #[]
+  deriving Inhabited
+
+/--
+Computes and caches the C3 linearization. Assumes parents have already been set with `setStructureParents`.
+If `relaxed` is false, then if the linearization cannot be computed, conflicts are recorded in the return value.
+-/
+partial def computeStructureResolutionOrder [Monad m] [MonadEnv m]
+    (structName : Name) (relaxed : Bool) : m StructureResolutionOrderResult := do
+  let env ← getEnv
+  if let some resOrder := getStructureResolutionOrder? env structName then
+    return { resolutionOrder := resOrder }
+  let parentNames := getStructureParentInfo env structName |>.map (·.structName)
+  -- Don't be strict about parents: if they were supposed to be checked, they were already checked.
+  let parentResOrders ← parentNames.mapM fun parentName => return (← computeStructureResolutionOrder parentName true).resolutionOrder
+
+  -- `resOrders` contains the resolution orders to merge.
+  -- The parent list is inserted as a pseudo resolution order to ensure immediate parents come out in order,
+  -- and it is added first to be the primary ordering constraint when there are ordering errors.
+  let mut resOrders := parentResOrders.insertAt 0 parentNames |>.filter (!·.isEmpty)
+
+  let mut resOrder : Array Name := #[structName]
+  let mut defects : Array StructureResolutionOrderConflict := #[]
+  -- Every iteration of the loop, the sum of the sizes of the arrays in `resOrders` decreases by at least one,
+  -- so it terminates.
+  while !resOrders.isEmpty do
+    let (good, name) ← selectParent resOrders
+
+    unless good || relaxed do
+      let conflicts := resOrders |>.filter (·[1:].any (· == name)) |>.map (·[0]!) |>.qsort Name.lt |>.eraseReps
+      defects := defects.push {
+        isDirectParent := parentNames.contains name
+        badParent := name
+        conflicts := conflicts.map fun c => (parentNames.contains c, c)
+      }
+
+    resOrder := resOrder.push name
+    resOrders := resOrders
+      |>.map (fun resOrder => resOrder.filter (· != name))
+      |>.filter (!·.isEmpty)
+
+  setStructureResolutionOrder structName resOrder
+  return { resolutionOrder := resOrder, conflicts := defects }
+where
+  selectParent (resOrders : Array (Array Name)) : m (Bool × Name) := do
+    -- Assumption: every resOrder is nonempty.
+    -- `n'` is for relaxation, to stop paying attention to end of `resOrders` when finding a good parent.
+    for n' in [0 : resOrders.size] do
+      let hi := resOrders.size - n'
+      for i in [0 : hi] do
+        let parent := resOrders[i]![0]!
+        let consistent resOrder := resOrder[1:].all (· != parent)
+        if resOrders[0:i].all consistent && resOrders[i+1:hi].all consistent then
+          return (n' == 0, parent)
+    -- unreachable, but correct default:
+    return (false, resOrders[0]![0]!)
+
+/--
+Gets the resolution order for a structure.
+-/
+def getStructureResolutionOrder [Monad m] [MonadEnv m]
+    (structName : Name) : m (Array Name) :=
+  (·.resolutionOrder) <$> computeStructureResolutionOrder structName (relaxed := true)
+
+/--
+Returns the transitive closure of all parent structures of the structure.
+This is the same as `Lean.getStructureResolutionOrder` but without including `structName`.
+-/
+partial def getAllParentStructures [Monad m] [MonadEnv m] (structName : Name) : m (Array Name) :=
+  (·.erase structName) <$> getStructureResolutionOrder structName
+
 end Lean
diff --git a/tests/lean/run/3467.lean b/tests/lean/run/3467.lean
new file mode 100644
index 000000000000..d3df62417278
--- /dev/null
+++ b/tests/lean/run/3467.lean
@@ -0,0 +1,139 @@
+import Lean
+/-!
+# Tests for structure resolution order.
+
+https://github.com/leanprover/lean4/issues/3467
+https://github.com/leanprover/lean4/issues/1881
+-/
+
+/-!
+Basic diamond
+-/
+
+set_option structure.strictResolutionOrder true
+set_option trace.Elab.structure.resolutionOrder true
+
+/-- info: [Elab.structure.resolutionOrder] computed resolution order: [A] -/
+#guard_msgs in structure A
+/-- info: [Elab.structure.resolutionOrder] computed resolution order: [B, A] -/
+#guard_msgs in structure B extends A
+/-- info: [Elab.structure.resolutionOrder] computed resolution order: [C, A] -/
+#guard_msgs in structure C extends A
+/-- info: [Elab.structure.resolutionOrder] computed resolution order: [D, B, C, A] -/
+#guard_msgs in structure D extends B, C
+
+def A.x (a : A) : Bool := default
+def B.x (b : B) : Nat := default
+def A.y (c : A) : Bool := default
+def C.y (d : C) : Nat := default
+
+variable (a : A) (b : B) (c : C) (d : D)
+
+/-- info: a.x : Bool -/
+#guard_msgs in #check a.x
+/-- info: b.x : Nat -/
+#guard_msgs in #check b.x
+/-- info: c.x : Bool -/
+#guard_msgs in #check c.x
+/-- info: d.x : Nat -/
+#guard_msgs in #check d.x
+/-- info: a.y : Bool -/
+#guard_msgs in #check a.y
+/-- info: b.y : Bool -/
+#guard_msgs in #check b.y
+/-- info: c.y : Nat -/
+#guard_msgs in #check c.y
+/-- info: d.toC.y : Nat -/
+#guard_msgs in #check d.y
+
+
+/-!
+Example resolution order failure
+-/
+
+/--
+warning: failed to compute strict resolution order:
+- parent 'B' must come after parent 'D'
+---
+info: [Elab.structure.resolutionOrder] computed resolution order: [D', B, D, C, A]
+-/
+#guard_msgs in
+structure D' extends B, D
+
+
+/-!
+Example from issue 3467.
+-/
+
+namespace Issue3467
+
+/-- info: [Elab.structure.resolutionOrder] computed resolution order: [Issue3467.X] -/
+#guard_msgs in
+structure X where
+  base : Nat
+
+/-- info: [Elab.structure.resolutionOrder] computed resolution order: [Issue3467.A, Issue3467.X] -/
+#guard_msgs in
+structure A extends X where
+  countA : Nat
+
+/-- info: [Elab.structure.resolutionOrder] computed resolution order: [Issue3467.B, Issue3467.X] -/
+#guard_msgs in
+structure B extends X where
+  countB : Nat
+
+namespace A
+
+def getTwiceCountA (a : A) := a.countA * 2
+
+end A
+
+namespace B
+
+def getTwiceCountB (b : B) := b.countB * 2
+
+end B
+
+/--
+info: [Elab.structure.resolutionOrder] computed resolution order: [Issue3467.C, Issue3467.A, Issue3467.B, Issue3467.X]
+-/
+#guard_msgs in
+structure C extends A, B
+
+def getCounts (c : C) :=
+  c.countA + c.countB
+
+def getTwiceCounts (c : C) :=
+  c.getTwiceCountA + c.getTwiceCountB
+--                     ^^^^ used to fail to resolve
+
+end Issue3467
+
+
+namespace Issue1881
+
+/-- info: [Elab.structure.resolutionOrder] computed resolution order: [Issue1881.Foo1] -/
+#guard_msgs in
+structure Foo1 where
+  a : Nat
+  b : Nat
+
+/-- info: [Elab.structure.resolutionOrder] computed resolution order: [Issue1881.Foo2] -/
+#guard_msgs in
+structure Foo2 where
+  a : Nat
+  c : Nat
+
+/--
+info: [Elab.structure.resolutionOrder] computed resolution order: [Issue1881.Foo3, Issue1881.Foo1, Issue1881.Foo2]
+-/
+#guard_msgs in
+structure Foo3 extends Foo1, Foo2 where
+  d : Nat
+
+def Foo1.bar1 (_ : Foo1) : Nat := 0
+def Foo2.bar2 (_ : Foo2) : Nat := 1
+example (x : Foo3) := x.bar1 -- works
+example (x : Foo3) := x.bar2 -- now works
+
+end Issue1881
diff --git a/tests/lean/run/structure.lean b/tests/lean/run/structure.lean
index 4731339e6f89..fc2432600519 100644
--- a/tests/lean/run/structure.lean
+++ b/tests/lean/run/structure.lean
@@ -28,7 +28,7 @@ info: #[const2ModIdx, constants, extensions, extraConstNames, header]
 #[toS2, toS1, x, y, z, toS3, w, s]
 (some [S4.toS2, S2.toS1])
 #[S2, S3]
-#[S2, S1, S3]
+#[S2, S3, S1]
 -/
 #guard_msgs in
 #eval show CoreM Unit from do
@@ -51,7 +51,7 @@ info: #[const2ModIdx, constants, extensions, extraConstNames, header]
   IO.println (getStructureFieldsFlattened env `S4)
   IO.println (getPathToBaseStructure? env `S1 `S4)
   IO.println (getStructureSubobjects env `S4)
-  IO.println (getAllParentStructures env `S4)
+  IO.println (← getAllParentStructures `S4)
   pure ()
 
 def dumpStructInfo (structName : Name) : CoreM Unit := do