Overwhelming evidence has accumulated in recent years that supernova explosions are intrinsically three-dimensional phenomena with significant departures from spherical symmetry. We review the evidence derived from spectropolarimetry that has established several key results: Virtually all supernovae are significantly aspherical near maximum light; core-collapse supernovae behave differently than thermonuclear (Type Ia) supernovae; the asphericity of core-collapse supernovae is more pronounced in the inner layers, showing that the explosion process is strongly aspherical; core-collapse supernovae tend to establish a preferred direction of asymmetry; and the asphericity is stronger in the outer layers of thermonuclear supernovae, providing constraints on the burning process. We emphasize the utility of the Q/U plane as a diagnostic tool and revisit SN 1987A and SN 1993J in a contemporary context. An axially symmetric geometry can explain many basic features of core-collapse supernovae, but significant departures from axial symmetry are needed to explain most events. We introduce a spectropolarimetry type to classify the range of behavior observed in polarized supernovae. Understanding asymmetries in supernovae is important for phenomena as diverse as the origins of gamma-ray bursts and the cosmological applications of Type Ia supernovae in studies of the dynamics of the universe.