In this article we consider a space B com G assembled from commuting elements in a Lie group G first defined by Adem, Cohen and Torres-Giese.We describe homotopy-theoretic properties of these spaces using homotopy colimits, and their role as a classifying space for transitionally commutative bundles.We prove that Z B com U is a loop space and define a notion of commutative K-theory for bundles over a finite complex X , which is isomorphic to OEX; Z B com U .We compute the rational cohomology of B com G for G equal to any of the classical groups SU.r/, U.q/ and Sp.k/, and exhibit the rational cohomologies of B com U , B com SU and B com