  The Euler characteristic is the only additive topological invariant for
spaces of certain sort, in particular, for manifolds with some finiteness
properties. A generalization of the notion of a manifold is the notion of a
V-manifold. Here we discuss a universal additive topological invariant of
V-manifolds: the universal Euler characterictic. It takes values in the ring
generated (as a Z-module) by isomorphism classes of finite groups. We also
consider the universal Euler characteristic on the class of locally closed
equivariant unions of cells in equivariant CW-complexes. We show that it is a
universal additive invariant satisfying a certain "induction relation". We give
Macdonald type equations for the universal Euler characteristic for V-manifolds
and for cell complexes of the described type.
