Birds are the only taxonomic group other than mammals that exhibit high-amplitude slow-waves in the electroencephalogram (EEG) during sleep. This defining feature of slow-wave sleep (SWS) apparently evolved independently in mammals and birds, as reptiles do not exhibit similar EEG activity during sleep. In mammals, the level of slow-wave activity (SWA) (low-frequency spectral power density) during SWS increases and decreases as a function of prior time spent awake and asleep, respectively, and therefore reflects homeostatically regulated sleep processes potentially tied to the function of SWS. Although birds also exhibit SWS, previous sleep deprivation studies in birds did not detect a compensatory increase in SWS-related SWA during recovery, as observed in similarly sleep-deprived mammals. This suggested that, unlike mammalian SWS, avian SWS is not homeostatically regulated, and therefore might serve a different function. However, we recently demonstrated that SWA during SWS increases in pigeons following short-term sleep deprivation. Herein we summarize research on avian sleep homeostasis, and cast our evidence for this phenomenon within the context of theories for the function of SWS in mammals. We propose that the convergent evolution of homeostatically regulated SWS in mammals and birds was directly linked to the convergent evolution of large, heavily interconnected brains capable of performing complex cognitive processes in each group. Specifically, as has been proposed for mammals, the interconnectivity that forms the basis of complex cognition in birds may also instantiate slow, synchronous network oscillations during SWS that in turn maintain interconnectivity and cognition at an optimal level.