Starting from a review on homology theory, we briefly survey some basic concepts and theoretical results on infinite dimensional Morse theory. To illustrate the applications of the theory, we discuss some classical applications of Morse theory to semilinear elliptic BVPs, quasilinear elliptic BVPs. Then, we study some Schrodinger type problems. The study of nonlinear Schrodinger-Poisson systems and quasilinear Schrodinger equations has been the focus of nonlinear analysis in the last two decades, almost all results in these problems require the Schrodinger operator to be positive. In this talk, as application of Morse theory, we will present our recent results on such problems with indefinite Schrodinger operator.