Over more than a century of research has established the fact that sleep benefits the retention of memory. In this review we aim to comprehensively cover the field of "sleep and memory" research by providing a historical perspective on concepts and a discussion of more recent key findings. Whereas initial theories posed a passive role for sleep enhancing memories by protecting them from interfering stimuli, current theories highlight an active role for sleep in which memories undergo a process of system consolidation during sleep. Whereas older research concentrated on the role of rapid-eye-movement (REM) sleep, recent work has revealed the importance of slow-wave sleep (SWS) for memory consolidation and also enlightened some of the underlying electrophysiological, neurochemical, and genetic mechanisms, as well as developmental aspects in these processes. Specifically, newer findings characterize sleep as a brain state optimizing memory consolidation, in opposition to the waking brain being optimized for encoding of memories. Consolidation originates from reactivation of recently encoded neuronal memory representations, which occur during SWS and transform respective representations for integration into long-term memory. Ensuing REM sleep may stabilize transformed memories. While elaborated with respect to hippocampus-dependent memories, the concept of an active redistribution of memory representations from networks serving as temporary store into long-term stores might hold also for non-hippocampus-dependent memory, and even for nonneuronal, i.e., immunological memories, giving rise to the idea that the offline consolidation of memory during sleep represents a principle of long-term memory formation established in quite different physiological systems.