The granule, CA1 and CA3 cells of the hippocampus have been much investigated during the last decade because there is superimposed on the standard features of synaptic transmission a very prolonged potentiation lasting for weeks that is called long-term potentiation. Evidently long-term potentiation is a promising candidate in the construction of a model for memory. The thesis here developed is that the influx of calcium ions across the membrane of the granule pyramidal cells plays the key role in the generation of long-term potentiation. The proposal makes it possible to account for the necessity of strong repetitive synaptic stimulation, preferably in bursts so as to optimize the conditions for the calcium influx. Studies on hippocampal slices with variations in the synaptic inputs in the granule cells give evidence of cooperativity, which is interpreted in relation to the threshold membrane depolarization for calcium influx. It is conjectured that the large increase of calcium in the granule and pyramidal cells results in the combination with the specific protein, calmodulin, to form a second messenger system, which produces metabolic changes leading to an increase in receptors of the postsynaptic membrane of the spine synapses, i.e. the postsynaptic densities, to the synaptic transmitter, glutamate. For example, Ca2+ could activate calcium-dependent kinases in the postsynaptic density resulting in the modification of protein components by phosphorylation. Other postsynaptic factors contributing to long-term potentiation are presumed to be protein synthesis with spine swelling and increased transport up the dendritic microtubules. There is discussion of the evidence for the alternative hypothesis that long-term potentiation is primarily presynaptic, being due to an increased output of transmitter. A unifying hypothesis is formulated, namely, that the primary event in long-term potentiation is in the increased sensitivity of the postsynaptic densities to the transmitter, and that, secondarily, this induces an increased output of transmitter from the presynaptic terminals by a trophic action across the synaptic cleft. It is shown how the proposed combination of calcium with calmodulin will account for the hypothesis of Marr that cognitive memory is due to conjunction potentiation. Furthermore, the Marr-Albus hypothesis for cerebellar learning is accounted for if the calcium-calmodulin messenger system causes the observed depression of the transmitter sensitivity of the spine synapses on Purkynĕ cells.