What are the units of attention? In addition to standard models holding that attention can select spatial regions and visual features, recent work suggests that in some cases attention can directly select discrete objects. This paper reviews the state of the art with regard to such 'object-based' attention, and explores how objects of attention relate to locations, reference frames, perceptual groups, surfaces, parts, and features. Also discussed are the dynamic aspects of objecthood, including the question of how attended objects are individuated in time, and the possibility of attending to simple dynamic motions and events. The final sections of this review generalize these issues beyond vision science, to other modalities and fields such as auditory objects of attention and the infant's 'object concept'.