Let GCM(d) denote the collection of groups (up to isomorphism) that appear as the torsion subgroup of a CM elliptic curve over a degree d number field. We completely determine GCM(d) for odd integers d and deduce a number of statistical theorems about the behavior of torsion subgroups of CM elliptic curves. Here are three examples: (1) For each odd d, the set of natural numbers d ′ with GCM(d ′ ) = GCM(d) possesses a well-defined, positive asymptotic density. (2) Let TCM(d) = max G∈G CM (d) #G; under the Generalized Riemann Hypothesis,