Birds exhibit two types of sleep that are in many respects similar to mammalian rapid eye movement (REM) and non-REM (NREM) sleep. As in mammals, several aspects of avian sleep can occur in a local manner within the brain. Electrophysiological evidence of NREM sleep occurring more deeply in one hemisphere, or only in one hemisphere - the latter being a phenomenon most pronounced in dolphins - was actually first described in birds. Such asymmetric or unihemispheric NREM sleep occurs with one eye open, enabling birds to visually monitor their environment for predators. Frigatebirds primarily engage in this form of sleep in flight, perhaps to avoid collisions with other birds. In addition to interhemispheric differences in NREM sleep intensity, the intensity of NREM sleep is homeostatically regulated in a local, use-depended manner within each hemisphere. Furthermore, the intensity and temporo-spatial distribution of NREM sleep-related slow waves varies across layers of the avian hyperpallium - a primary visual area - with the slow waves occurring first in, and propagating through and outward from, thalamic input layers. Slow waves also have the greatest amplitude in these layers. Although most research has focused on NREM sleep, there are also local aspects to avian REM sleep. REM sleep-related reductions in skeletal muscle tone appear largely restricted to muscles involved in maintaining head posture. Other local aspects of sleep manifest as a mixture of features of NREM and REM sleep occurring simultaneously in different parts of the neuroaxis. Like monotreme mammals, ostriches often exhibit brainstem-mediated features of REM sleep (muscle atonia and REMs) while the hyperpallium shows EEG slow waves typical of NREM sleep. Finally, although mice show slow waves in thalamic input layers of primary sensory cortices during REM sleep, this is not the case in the hyperpallium of pigeons, suggesting that this phenomenon is not a universal feature of REM sleep. Collectively, the local aspects of sleep described in birds and mammals reveal that wakefulness, NREM sleep, and REM sleep are not always discrete states.