Convex polytopes have attracted human attention since ancient times. Euler's formula, V-E+F=2, for the numbers of vertices, edges, and faces F of a 3-dimensional polytope, is among the most important landmarks of mathematics, and it is a starting point for a rich theory of face numbers of polytopes in high dimensions. Another important landmark is Cauchy's rigidity theorem. In the lecture I will present a few major combinatorial results about polytopes, some connections to other areas of mathematics, pure and applied, and fascinating open problems.