For positive integers k,n, we investigate the simplicial complex NM_k(n)\mathsf{NM}_{k}(n) of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy equivalent to a wedge of spheres. The number and dimension of the spheres in the wedge are determined, and (partially conjectural) links to other combinatorially defined complexes are described. In addition we study for positive integers r,s and k the simplicial complex BNM_k(r,s)\mathsf{BNM}_{k}(r,s) of all bipartite graphs G on bipartition [r]È[[`(s)]][r]\cup [\bar{s}] such that there is no matching of size k in G, and obtain results similar to those obtained for NM_k(n)\mathsf{NM}_{k}(n) .