Dendritic spines are among the smallest structures in the brain, yet they hold some of the deepest clues to how neurons compute and store memories. In this lecture, I will revisit the classical question of spine biophysics in the new age of connectomics, where nanometer-resolution electron microscopy allows us to reconstruct whole neurons with thousands of synapses and spines. I will show how spine density, spine-neck resistance, and the irregular geometry of individual spines can shape local synaptic signals, generate fast voltage dynamics, influence excitatory/inhibitory gating, and affect communication between neighboring spines. I will then discuss how automated reconstruction pipelines and biophysical modeling now make it possible to build digital neurons at spine and synapse resolution, and how AI-based approaches, such as our new AI-based TwinProp algorithm, allow us to ask - and get answer - to a fundamental question: what can a neuron compute? I will end by claiming that we now enter a new age of connectomics that provides an essential constraint on understanding how synapses, neurons, and circuits perform their computational functions.