Richard Stanley's resolution of McMullen's g-conjecture in the 1970s using the cohomology of toric varieties sparked the use of toric geometry in geometric combinatorics. Since then, combinatorialists have looked to toric varieties and toric ideals as a source of questions and techniques for reasoning about polytopes. We take the former point of view and describe the correspondence between this large class of (normal, complex) varieties and the combinatorial data of polytopes. In particular, we discuss how the combinatorics of polytopes and polytopal fans not only characterizes all toric varieties, but also encodes a wealth of intrinsic geometric information about the variety itself. Time-permitting, we discuss some of the aforementioned combinatorial applications of toric geometry (and adjacent ideas).
(Based on Toric Varieties by Cox, Little, and Schenck.)