This article begins with a description of three notions of compositions that have parts in a fixed set of positive integers: linear compositions, circular compositions, and cyclic compositions. We describe some relations among these notions and review generating functions to count each type of composition. The main result of the paper generalizes these known results to count compositions in which the number of parts is required to be congruent to q modulo m for some fixed 0 ≤ q ≤ m − 1. The particular case m = 2, q = 1 yields the compositions with an odd number of parts. The latter sections apply the main theorem to several special cases, including compositions in which the parts are allowed to be drawn from a multiset.