Morse theory was originally due to Marston Morse [5]. It gives us a method to study the topology of a manifold using the information of the critical points of a Morse function defined on the manifold. Based on the same idea, Morse homology was introduced by Thom, Smale, Milnor, and Witten in various forms. This paper is a survey of some work in this direction. The first part of the paper focuse...