It is well known that the number of distinct non-crossing matchings of n half-circles in the half-plane with endpoints on the x-axis equals the n^th Catalan number C_n. This paper generalizes that notion of linear non-crossing matchings, as well as the circular non-crossing matchings of Goldbach and Tijdeman, to non-crossings matchings of curves embedded within an annulus. We prove that the number of such matchings | Ann(n, m) | with n exterior endpoints and m interior endpoints correspond to an entirely new, one-parameter generalization of the Catalan numbers with C_n = | Ann(2n + 1, 1) |. We also develop bijections between specific classes of annular non-crossing matchings and other combinatorial objects such as binary combinatorial necklaces and planar graphs. Finally, we use Burnside's lemma to obtain an explicit formula for | Ann(n, m) | for all integers n, m ≥ 0.