Here we consider a matching model where agents are heterogeneous and util-ities nontransferable. We utilize this framework to study how equilibrium sorting takes place in marriage markets. We impose conditions that guarantee the exis-tence of a steady state equilibrium and then characterize it. Several examples are developed to illustrate the richness of equilibria. The model reveals an interesting sorting externality that can support multiple steady state equilibria, even with constant returns to matching. The matching framework, developed by Mortensen [1982, 1985], Diamond [1982], and Pissarides [1990], has proved to be a useful tool in labor economics, macroeconomics, and monetary theory. Here we consider a matching model where agents ar...