In an isolated physical system, one would expect (physically) the total energy of that system to be positive. In the general relativity setting, for example, we may näively suspect that the total energy of some space-like hypersuface (a choice of space for a fixed time) to be simply the sum of the energy associated to the matter field(s) and the energy associated to the gravitational field. And surely in the abscence of such fields the total energy should be zero. The situation, however, is complicated by the coupling of the matter fields and gravitational fields as prescribed by Eistein’s field equations in general relativity: the energy associated to the matter field and the energy associated to the gravitational field cannot be defined seperately. Nevertheless, the Positive Energy Theorem, first proved by Schoen and Yau , and later by Witten , reassures our intuition through mathematics. The result and its proofs are non-trivial, and as we shall see, the non-triviality starts with the mathematical formulation of the intuitively simple problem. In this paper, we study the mathematical formulation of the problem, and some general aspects of Schoen and Yau’s proof as outlined in a paper by J. Kazdan . The four main ingredients required to understand the mathematical formulation of the Positive Energy Theomem are as follows.