  This paper announces results on the behavior of some important algebraic and
topological invariants --- Euler characteristic, arithmetic genus, and their
intersection homology analogues; the signature, etc. --- and their associated
characteristic classes, under morphisms of projective algebraic varieties. The
formulas obtained relate global invariants to singularities of general complex
algebraic (or analytic) maps. These results, new even for complex manifolds,
are applied to obtain a version of Grothendieck-Riemann-Roch, a calculation of
Todd classes of toric varieties, and an explicit formula for the number of
integral points in a polytope in Euclidean space with integral vertices.
