The course starts with an introduction to the basic theory in second order parabolic equations, including a brief review of the second order elliptic theory, then discusses basic theory about the heat equation, including fundamental solution, Schauder and Lp estimates, maximal principle, Harnack inequality for parabolic equations first on Euclidean space then on manifolds. Later we cover aspects of the Hamilton Ricci Flow on manifolds, including the study of Perelman's W-functional; application to the 'pinching results' along the Ricci Flow for problems in conformal geometry; and recent works of Gursky, Chang-Gursky-S. Zhang, et al.