We use entropy methods to prove asymptotic upper bounds for the number of matchings of a given size l in a graph G with a given degree sequence. In particular, for a d-regular, N-vertices graph G, our bound is best possible up to an error factor exp(o(N)). This represents the best progress to date on the "Upper Matching Conjecture" of Friedland, Krop, Lundow and Markstrom. Joint with Jeff Kahn.