In this lecture we will focus on one of the most beautiful concept of Mathematics; that is, Topology and Cohomology. We will need the knowledge of differential forms from which we will start by recalling some of the important things that we have already discussed in depth on differential forms. We will talk about closed and exact forms and the relation in between them. We shall see that it is the closed but not exact forms which provides us with useful topological information about the space. This is where we will introduce the quotient vector space which I will refer to as the P-th Cohomology of the manifold and will first come across what is called the P-th Betti Number. I will continue with some examples involving calculations where we consider various types of Cohomology and see the importance of the results where I define the Poincare Duality Theorem. At the end I will mention why all of this is useful to us as Physicists where I mention one of the use in String Theory.