To every finite collection of graphs that is closed under removal of edges, one can associate an abstract simplicial complex whose faces are the edge sets of the graphs in the collection. Graph complexes have provided an important link between combinatorics and algebra, topology and geometry. In this talk we will discuss the simplicial complex associated with the collection of subgraphs of a graph G whose maximum vertex degree is at most b. Some special cases which have arisen in various contexts in the recent literature are the matching complex (G is a complete graph and b=1) and the chessboard complex (G is a complete bipartite graph and b=1). Topological properties of the matching complex were first investigated by Bouc in connection with Quillen complexes, and topological properties of the chessboard complex were first investigated by Garst in connection with Coxeter complexes. These complexes also have arisen in connection with homology of nilpotent Lie algebras, and in connection with free resolutions of certain modules over quotients of polynomial rings. I will present some recent results on the homology of bounded degree graph complexes, and describe some of the combinatorial techniques employed. In particular, the representation of the symmetric group on complex homology and torsion in integral homology will be discussed.