EEG slow waves are the epitome of deep nonREM sleep. The level of slow-wave activity (SWA; defined as spectral power in the 0.5-4.5 Hz band) in the initial part of sleep is determined by prior sleep and waking, and thereby represents a marker of a homeostatic sleep regulating process (Process S). Models based on SWA were successful in simulating sleep architecture in a variety of experimental protocols. SWA is an exceptional sleep variable in that it is little influenced by circadian phase and variations of the photoperiod. There is recent evidence that it is not waking per se but the absence of sleep, which engenders a rise in sleep propensity. Thus animals emerging from the hypometabolic states of hibernation or daily torpor exhibit an increase in SWA akin to sleep deprivation. Recent human studies showed SWA to be a marker of a local, use-dependent facet of sleep. Selective activation of specific cortical areas during waking enhanced SWA over the activated region during sleep. A frontal predominance of power in the 2-Hz band was documented in the initial part of a normal sleep episode. Sleep homeostasis may be a valuable concept for exploring the evolutionary origin of sleep. Thus 'rest homeostasis' has been demonstrated in invertebrate species, and the search for homologies of rest and sleep on a molecular genetic level has begun. Conceptualizing and characterizing sleep as a regulated process may eventually shed light on its function.