Equiangularity (also called max-min angle criterion) is a well-known property of some planar triangulations that refine the Delaunay diagram. In this paper we generalize the notion of equiangularity to decompositions in inscribable polygons and we show that it characterizes the planar Delaunay diagram, even if more than three sites are cocircular. This result does not extend to higher dimensions. However, we characterize the Delaunay diagram in any dimension by a kind of dual property that we prove both with line angles and with solid angles. We also establish a local equiangularity of Delaunay diagrams in any dimension, and an angular characterization of self-centered diagrams. Finally, we show that these angular properties can, when appropriately defined, be generalized to the farthest point Delaunay diagram.