The Centerpoint Theorem states that, for any set S of n points in ℝ^d, there exists a point p in ℝ^d such that every closed halfspace containing p contains at least n/(d+1) points of S. We consider generalizations of the Centerpoint Theorem in which halfspaces are replaced with wedges (cones) of angle α. In ℝ², we give bounds that are tight for all values of α and give an O(n) time algorithm to find a point satisfying these bounds. We also give partial results for ℝ³ and, more generally, ℝ^d.