It is well known that every bipartite graph with vertex classes of size n whose minimum degree is at least n/2 contains a perfect matching. We prove an analog of this result for hypergraphs. We also prove several related results that guarantee the existence of almost perfect matchings in r-uniform hypergraphs of large minimum degree. Our bounds on the minimum degree are essentially best possible. (C) 2005 Wiley Periodicals, Inc.