Although long-term memory consolidation is supported by sleep, it is unclear how it differs from that during wakefulness. Our review, focusing on recent advances in the field, identifies the repeated replay of neuronal firing patterns as a basic mechanism triggering consolidation during sleep and wakefulness. During sleep, memory replay occurs during slow-wave sleep (SWS) in hippocampal assemblies together with ripples, thalamic spindles, neocortical slow oscillations, and noradrenergic activity. Here, hippocampal replay likely favors the transformation of hippocampus-dependent episodic memory into schema-like neocortical memory. REM sleep following SWS might balance local synaptic rescaling accompanying memory transformation with a sleep-dependent homeostatic process of global synaptic renormalization. Sleep-dependent memory transformation is intensified during early development despite the immaturity of the hippocampus. Overall, beyond its greater efficacy, sleep consolidation differs from wake consolidation mainly in that it is supported, rather than impaired, by spontaneous hippocampal replay activity possibly gating memory formation in neocortex.