Synapses are diverse in form and function. While there are strong evidential and theoretical reasons for believing that memories are stored at synapses, the concept of a specialized “memory synapse” is rarely discussed. Here, we review the evidence that memories are stored at the synapse and consider the opposing possibilities. We argue that if memories are stored in an active fashion at synapses, then these memory synapses must have distinct molecular complexes that distinguish them from other synapses. In particular, examples from Aplysia sensory-motor neuron synapses and synapses on defined engram neurons in rodent models are discussed. Specific hypotheses for molecular complexes that define memory synapses are presented, including persistently active kinases, transmitter receptor complexes and trans-synaptic adhesion proteins.