  We define a Grothendieck ring of varieties with finite groups actions and
show that the orbifold Euler characteristic and the Euler characteristics of
higher orders can be defined as homomorphisms from this ring to the ring of
integers. We describe two natural $\lambda$-structures on the ring and the
corresponding power structures over it and show that one of these power
structures is effective. We define a Grothendieck ring of varieties with
equivariant vector bundles and show that the generalized ("motivic") Euler
characteristics of higher orders can be defined as homomorphisms from this ring
to the Grothendieck ring of varieties extended by powers of the class of the
complex affine line. We give an analogue of the Macdonald type formula for the
generating series of the generalized higher order Euler characteristics of
wreath products.
