This will be an introductory course on the combinatorial theory of convex polytopes and the enumerative theory of partially ordered sets (posets). The plan is to develop linear inequality theory enough to develop the geometric structure of polyhedral convex sets, and from this to study their combinatorial structure. Of particular interest will be f-vectors of convex polytopes. To study these combinatorial properties, we will consider the lattice of faces of a convex polytope and its Möbius function, and more generally the class of Eulerian posets. No previous experience with anything but linear algebra will be assumed.