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We study the C*-algebra crossed product $C_0(X)\rtimes G$ of a locallycompact group $G$ acting properly on a locally compact Hausdorff space $X$.Under some mild extra conditions, which are automatic if $G$ is discrete or aLie group, we describe in detail, and in terms of the action, the primitiveideal space of such crossed products as a topological space, in particular withrespect to its fibring over the quotient space $G\backslash X$. We also givesome results on the $\K$-theory of such C*-algebras. These more or less computethe $\K$-theory in the case of isolated orbits with non-trivial (finite)stabilizers. We also give a purely $\K$-theoretic proof of a result due to PaulBaum and Alain Connes on (\K)-theory with complex coefficients of crossedproducts by finite groups.

Structure and K-theory of crossed products by proper actions