The Hilbert series of a simplicial complex S (i.e., of the associated Stanley-Reisner ring) counts how many monomials supported on faces of S exist in each possible degree. The Ehrhart series of a lattice polytope P is a combinatorial gadget that counts the number of lattice points of P and of its dilations. In this introductory seminar we will investigate the interplay between the combinatorial and the algebraic aspects of such generating functions, including how the two constructions can sometimes come together in special cases. We will then start moving towards an equivariant setting, where the given simplicial complex or lattice polytope additionally carries a finite group action.