Morse Theory lets us use tools of calculus to understand the topology of a manifold. We will introduce the basic ideas and motivations of Morse theory, explaining how the homotopy type of a manifold M is determined by nondegenerate critical points of a function f on M. We then apply these tools to the energy functional, allowing us to determine the homotopy type of the path space of a complete Riemannian manifold. Milnor calls this “The Fundamental Theorem of Morse Theory”; as time allows, we will sketch the proof.