A growing body of evidence indicates that sleep promotes memory consolidation. Although the first experimental evidence for this positive influence of sleep on memory was collected more than a century ago, the potential underlying neural mechanisms begins only to be conceptualized and experimentally characterized. A first hypothesis contrasted the influence of non-rapid eye movement (NREM) sleep and rapid eye movement (REM) sleep on declarative and procedural memories, respectively. As the understanding of the effects of sleep on memory consolidation during sleep progressed, the hypotheses were increasingly framed in terms of neural processes occurring with NREM and REM sleep, especially associated with phasic events such as slow waves, spindles or phasic REM sleep. This paper reviews two of these hypotheses: the synaptic downscaling and the systemic consolidation during non NREM sleep.