Euler’s formula, V−E+F= 2, places restrictions on the set of all vectors (V, E, F) that can appear when counting the number of vertices, edges, and faces in a planar graph. Similarly, the orientability and Euler characteristic of a surface completely characterize the combinatorial structure of any tri-angulation of that surface. In this talk, I will discuss generalizations of these results to triangulations of higher-dimensional manifolds. The goal will be to address the question: “What can be said about the number of vertices, edges, triangles, etc., in a triangulated d-manifold?” Along the way, I will introduce some simple yet powerful algebraic tools that can be used to help answer this question that is rooted in combinatorial topology.