  We propose a definition of what should be meant by a {\it proper} action of a
locally compact group on a C*-algebra. We show that when the C*-algebra is
commutative this definition exactly captures the usual notion of a proper
action on a locally compact space. We then discuss how one might define a {\it
generalized fixed-point algebra}. The goal is to show that the generalized
fixed-point algebra is strongly Morita equivalent to an ideal in the crossed
product algebra, as happens in the commutative case. We show that one candidate
gives the desired algebra when the C*-algebra is commutative. But very recently
Exel has shown that this candidate is too big in general. Finally, we consider
in detail the application of these ideas to actions of a locally compact group
on the algebra of compact operators (necessarily coming from unitary
representations), and show that this gives an attractive view of the subject of
square-integrable representations.

Operator Algebras;Representation Theory