This paper studies matching in vertical networks, generalizing the theory of matching in two-sided markets. It gives sufficient conditions for the existence of stable networks and presents an algorithm for finding two of them. One is the best stable network for the agents on the “upstream” end of an industry. The other is best for the agents on the “downstream” end. The paper describes several properties of the set of stable networks and discusses applications of the theory to the design of matching markets with more than two types of agents and to the empirical analysis of supply chains. The paper is available at http://faculty-gsb.stanford.edu/ostrovsky/papers/chains.pdf.