Sleep is crucial for healthy cognition, including memory. The two main phases of sleep, REM (rapid eye movement) and non-REM sleep, are associated with characteristic electrophysiological patterns that are recorded using surface and intracranial electrodes. These patterns include sharp-wave ripples, cortical slow oscillations, delta waves, and spindles during non-REM sleep and theta oscillations during REM sleep. They reflect the precisely timed activity of underlying neural circuits. Here, we review how these electrical signatures have been guiding our understanding of the circuits and processes sustaining memory consolidation during sleep, focusing on hippocampal theta oscillations and sharp-wave ripples and how they coordinate with cortical patterns. Finally, we highlight how these brain patterns could also sustain sleep-dependent homeostatic processes and evoke several potential future directions for research on the memory function of sleep.