I will discuss three classes examples in which the geometry of an algebraic variety illuminates a certain combinatorial object. The first example will involve the classical relationship between toric varieties and polytopes. The next two examples will deal with a relatively new class of spaces called hypertoric varieties. These may be thought of as quaternionifications of toric varieties, and they interact richly with the combinatorics of matroids, or of finite collections of hyperplanes in a vector space.