  We define the equivariant Chern-Schwartz-MacPherson class of a possibly
singular algebraic variety with a group action over the complex number field
(or a field of characteristic 0). In fact, we construct a natural
transformation from the equivariant constructible function functor to the
equivariant homology functor (in the sense of Totaro-Edidin-Graham), which may
be regarded as MacPherson's transformation for (certain) quotient stacks. We
discuss on other type Chern/Segre classes and give some applications
generalizing orbifold Euler characteristics and Thom polynomials of
singularities. The Verdier-Riemann-Roch formula takes a key role throughout.
