In this course we will study multivariate hyper-geometric functions in the sense of Gel’f and, Kapranov, and Zelevinsky (GKZ systems). These functions generalize the classical hyper-geometric functions of Gauss, Horn, Appell, and Lauricella. We will emphasize the algebraic methods of Saito, Sturmfels, and Takayama to construct hyper-geometric series and the connection with deformation techniques in commutative algebra. We end with a brief discussion of the classification problem for rational hyper-geometric functions.