Keller-Segel　systems　in　two　and　three　space　dimensions　with　an　additional　cross-diffusion　term　in　the　equation　for　the　chemical　concentration　are　analyzed.　The　cross-diffusion　term　has　a　stabilizing　effect　and　leads　to　the　global-in-time　existence　of　weak　solutions.　The　limit　of　vanishing　cross-diffusion　parameter　is　proved　rigorously　in　the　parabolic-elliptic　and　parabolic-parabolic　cases.　When　the　signal　production　is　sublinear,　the　existence　of　global-in-time　weak　solutions　as　well　as　the　convergence　of　the　solutions　to　those　of　the　classical　parabolic-elliptic　Keller-Segel　equations　are　proved.　The　proof　is　based　on　a　reformulation　of　the　equations　eliminating　the　additional　cross-diffusion　term　but　making　the　equation　for　the　cell　density　quasilinear.　For　superlinear　signal　production　terms,　convergence　rates　in　the　cross-diffusion　parameter　are　proved　for　local-in-time　smooth　solutions　(since　finite-time　blow　up　is　possible).　The　proof　is　based　on　careful　H-s(Omega)　estimates　and　a　variant　of　the　Gronwall　lemma.　Numerical　experiments　in　two　space　dimensions　illustrate　the　theoretical　results　and　quantify　the　shape　of　the　cell　aggregation　bumps　as　a　function　of　the　cross-diffusion　parameter.　(C)　2019　Elsevier　Ltd.　All　rights　reserved.