  We present seven theorems on the structure of prime order torsion points on
CM elliptic curves defined over number fields. The first three results refine
bounds of Silverberg and Prasad-Yogananda by taking into account the class
number of the CM order and the splitting of the prime in the CM field. In many
cases we can show that our refined bounds are optimal or asymptotically
optimal. We also derive asymptotic upper and lower bounds on the least degree
of a CM-point on X_1(N). Upon comparison to bounds for the least degree for
which there exist infinitely many rational points on X_1(N), we deduce that,
for sufficiently large N, X_1(N) will have a rational CM point of degree
smaller than the degrees of at least all but finitely many non-CM points.
