Integral cohomology of singular toric varieties

The singular cohomology and Chow cohomology, with Q-coefficients, of projective toric varieties with at worst orbifold singularities are well-understood, but interesting problems remain for toric varieties with more serious singularities and for cohomology with Z-coefficients. I will present a computation of the equivariant Chow cohomology of singular toric varieties with Z-coefficients in terms of piecewise-polynomial functions on fans, using Kimura's inductive methods with envelopes and resolutions of singularities. As time premits, I will also discuss some open questions and conjectures about the singular cohomology of toric varieties and its relation to Chow cohomology, and about the cohomology of real toric varieties.