The `chessboard complex' \(\Delta_{m,n}\) is the abstract simplicial complex given by all non-taking rook configurations on an \((m \times n)\)- chessboard. This complex is the fundamental structure behind quite distinct problems: it appears as the matching complex of a complete bipartite graph \(K_{m,n}\), a coset complex in the symmetric group, etc. In several applications crucial data are given by connectivity properties of chessboard complexes. The main result is that \(\Delta_{m,n}\) is homotopically \((\nu-2)\)-connected, for \(\nu:=\min \{ \lfloor (m+n+1)/3) \rfloor,m,n\}\). This bound is conjectured to be sharp. Similar lower bounds for connectivity are proved for the chessboard complexes of higher-dimensional chessboards, and for the matching complexes of complete hypergraphs. The proofs rely on a sharpened connectivity version of the nerve lemma --- a versatile tool that should have other applications as well. The paper also gives a detailed analysis of some small chessboard complexes, surprisingly exhibiting torsion in homology, such that the depth of the Stanley Reisner rings in some cases depends on the characteristic of the field. Shellability of the \((\nu-1)\)-skeleta of chessboard complexes, implying \((\nu-2)\)-connectivity, was subsequently proved by the reviewer [Shellability of chessboard complexes, Israel J. Math. (1994), to appear].