diff --git a/README.md b/README.md
index b9695a5..b63039a 100644
--- a/README.md
+++ b/README.md
@@ -19,8 +19,10 @@ Each `Challenge` has a `Component` as an attribute, and also has a target that c
 
 ## Example
 ```python
+# Select the challenge.
 challenge = invrs_gym.challenges.ceviche_lightweight_waveguide_bend()
 
+# Define loss function, which also returns auxilliary quantities.
 def loss_fn(params):
     response, aux = challenge.component.response(params)
     loss = challenge.loss(response)
@@ -30,12 +32,15 @@ def loss_fn(params):
 
 value_and_grad_fn = jax.value_and_grad(loss_fn, has_aux=True)
 
+# Select an optimizer.
 opt = invrs_opt.density_lbfgsb(beta=4)
 
+# Generate initial parameters, and use these to initialize the optimizer state.
 params = challenge.component.init(jax.random.PRNGKey(0))
 state = opt.init(params)
 
-for _ in range(steps):
+# Carry out the optimization.
+for i in range(steps):
     params = opt.params(state)
     (value, (response, distance, aux)), grad = value_and_grad_fn(params)
     state = opt.update(grad=grad, value=value, params=params, state=state)
diff --git a/notebooks/readme_example.ipynb b/notebooks/readme_example.ipynb
index 7a5de06..8977693 100644
--- a/notebooks/readme_example.ipynb
+++ b/notebooks/readme_example.ipynb
@@ -28,7 +28,7 @@
    "source": [
     "challenge = invrs_gym.challenges.ceviche_lightweight_waveguide_bend()\n",
     "\n",
-    "\n",
+    "# Define loss function, which also returns auxilliary quantities.\n",
     "def loss_fn(params):\n",
     "    response, aux = challenge.component.response(params)\n",
     "    loss = challenge.loss(response)\n",
@@ -36,14 +36,16 @@
     "    metrics = challenge.metrics(response, params, aux)\n",
     "    return loss, (response, distance, aux)\n",
     "\n",
-    "\n",
     "value_and_grad_fn = jax.value_and_grad(loss_fn, has_aux=True)\n",
     "\n",
+    "# Select an optimizer.\n",
     "opt = invrs_opt.density_lbfgsb(beta=4)\n",
     "\n",
+    "# Generate initial parameters, and use these to initialize the optimizer state.\n",
     "params = challenge.component.init(jax.random.PRNGKey(0))\n",
     "state = opt.init(params)\n",
     "\n",
+    "# Carry out the optimization.\n",
     "data = []\n",
     "for i in range(36):\n",
     "    params = opt.params(state)\n",
@@ -59,6 +61,7 @@
    "metadata": {},
    "outputs": [],
    "source": [
+    "# Create an animated gif showing the evolution of the waveguide bend design.\n",
     "anim = gifcm.AnimatedFigure(figure=plt.figure(figsize=(8, 4)))\n",
     "\n",
     "for i, _, params, aux in data:\n",