We will give an introduction to the notion of A-hypergeometric systems due to Gel'fand, Kapranov and Zelevinsky, and their solutions, the A- hypergeometric functions. These are multivariate generalizations of classical hypergeometric equations and functions with very beautiful combinatorial and algebro-geometric ingredients. We will also survey some recent results obtained in collaboration with Cattani, Matusevich, Sturmfels, and Sadykov.