Baleen whales are gigantic obligate filter feeders that exploit aggregations of small-bodied prey in littoral, epipelagic, and mesopelagic ecosystems. At the extreme of maximum body size observed among mammals, baleen whales exhibit a unique combination of high overall energetic demands and low mass-specific metabolic rates. As a result, most baleen whale species have evolved filter-feeding mechanisms and foraging strategies that take advantage of seasonally abundant yet patchily and ephemerally distributed prey resources. New methodologies consisting of multi-sensor tags, active acoustic prey mapping, and hydrodynamic modeling have revolutionized our ability to study the physiology and ecology of baleen whale feeding mechanisms. Here, we review the current state of the field by exploring several hypotheses that aim to explain how baleen whales feed. Despite significant advances, major questions remain about the processes that underlie these extreme feeding mechanisms, which enabled the evolution of the largest animals of all time.