Abstract: The aim of this talk is to provide an introduction to Morse and Morse-Bott theory. A Morse-Bott function on a manifold is a smooth function whose critical set is a closed submanifold with non-degenerate Hessian in the normal direction. The critical manifold and the trajectory spaces connecting them give rise to a complex. In the Morse case, i.e. the case when the critical points are isolated, this complex is the vector space generated by the critical points; and the differential is obtained by counting instantons, the isolated trajectories connecting critical points. We will present a proof of the fact that the Morse-Bott complex computes the cohomology of the underlying manifold. The rest of the talk will be devoted to applications of Morse and Morse-Bott theory.