  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-crossings matchings of Goldbach and
Tijdeman, to non-crossings matchings of $n$ line segments embedded within an
annulus. We prove that the number of such matchings $\vert Ann(n,m) \vert$ with
$n$ exterior endpoints and $m$ interior endpoints correspond to an entirely
new, one-parameter generalization of the Catalan numbers with $C_n = \vert
Ann(1,m) \vert$. 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 $\vert Ann(n,m) \vert$ for all $n,m \geq 0$.
