There are deep and subtle relations between the numbers of vertices, edges, triangles, and higher dimensional faces of a polytope. The so-called g-theorem makes these precise by giving necessary and sufficient conditions for an integer vector to be counting the faces of a polytope. In this introductory talk, we'll discuss Stanley's ingenious proof of the g-theorem, which invokes the topology of algebraic varieties. This will take us into the realm of toric varieties, where combinatorics and algebraic geometry intertwine. This bird's-eye view will not include hard algebraic geometry—interesting details will be the subjects of future meetings of the learning seminar.