  In this paper we develop an equivariant intersection theory for actions of
algebraic groups on algebraic schemes. The theory is based on our construction
of equivariant Chow groups. They are algebraic analogues of equivariant
cohomology groups which satsify the formal properties of ordinary Chow groups.
In addition, they enjoy many of the properties of equivariant cohomology. The
principal results are: (1) We prove the existence of canonical intersection
products on the Chow groups of geometric quotients of smooth varieties- even
when the stabilizers of geometric points are non-reduced. (2) We construct a
Todd class map from equivariant $K$-theory of coherent sheaves to a completion
of equivariant Chow groups, and prove that a completion of equivariant
$K$-theory is isomorphic to the completion of equivariant Chow groups. (3) We
prove a localization theorem for torus actions and use it to give a
characteristic free proof of the Bott residue formula for actions of tori on
complete smooth varieties.
