we begin by surveying some highlights of the theory of increasing and decreasing subsequences of permutations, including (1) connections with young tableaux and the $rsk$ algorithm, (2) the expected length is($w$) of the longest increasing subsequence of a random permutation $w$ of $1$,$2$,...,n, and (3) the limiting distribution of is($w$) due to baik, deift, and johansson. we will then discuss how these results carry over to (complete) matchings $m$ on the vertices $1$,$2$,...,$2n$. the analogue of increasing/decreasing subsequences will be shown to be related to crossings and nestings of $m$.