Toric varieties are of interest both in their own right as algebraic varieties, and for their application to the theory of convex polytopes. For instance, Danilov used the Hirzebruch-Riemann-Roch theorem to establish a direct connection between the problem of counting the number of lattice points in a convex polytope and the Todd classes of toric varieties. In this talk, I will discuss the computation of generalized homology Todd classes of (possibly singular) toric varieties, with applications to weighted lattice point counting. In the simplicial context, I will present a formula which computes these homology classes in terms of (cohomological) "mock" characteristic classes plus explicit contributions of the singular locus. This is joint work with Joerg Schuermann.