  A perfect matching M in an edge-colored complete bipartite graph K_{n,n} is
rainbow if no pair of edges in M have the same color. We obtain asymptotic
enumeration results for the number of rainbow matchings in terms of the maximum
number of occurrences of a color. We also consider two natural models of random
edge-colored K_{n,n} and show that, if the number of colors is at least n, then
there is with high probability a random matching. This in particular shows that
almost every square matrix of order n in which every entry appears at most n
times has a Latin transversal.
