Abstract: I will explain how complex and real toric varieties and their non-negative parts can easily be defined topologically. This gives in particular canonical cell decompositions of these spaces.

I will also discuss consequences to the ordinary and equivariant integral cohomology of toric varieties. For example, if the ordinary cohomology is concentrated in even degrees, then the equivariant cohomology can be described by piecewise polynomials. If the toric variety is in addition smooth or compact, then its ordinary cohomology is necessarily torsion-free.