Dendritic spines, which receive most of the excitatory synaptic input in the cerebral cortex, are heterogeneous with regard to their structure, stability and function. Spines with large heads are stable, express large numbers of AMPA-type glutamate receptors, and contribute to strong synaptic connections. By contrast, spines with small heads are motile and unstable and contribute to weak or silent synaptic connections. Their structure-stability-function relationships suggest that large and small spines are "memory spines" and "learning spines", respectively. Given that turnover of glutamate receptors is rapid, spine structure and the underlying organization of the actin cytoskeleton are likely to be major determinants of fast synaptic transmission and, therefore, are likely to provide a physical basis for memory in cortical neuronal networks. Characterization of supramolecular complexes responsible for synaptic memory and learning is key to the understanding of brain function and disease.