In this talk I will survey some results on statistical properties of matchings of very large and infinite graphs. The main goal of the talk is to describe a few applications of a new concept called matching measure. These applications include new results on the number of (perfect) matchings in large girth graphs as well as simple new proofs of certain statistical physical theorems. This is joint work with various subsets of Miklós Abért, Péter E. Frenkel, Tamás Hubai and Gábor Kun.