The Ricci flow which was invented by Richard Hamilton in his 1982 paper, is now a basic concept in differential geometry, and “a must reckoned with” tool in geometric analysis. On the other hand, the Li-Yau Harnack inequality which is an inequality developed to estimate solution to the heat equation on a manifold with nonnegatitive Ricci tensor. The major aim of this course is to introduce the basics of Hamilton’s Ricci flow and the theory of famous Li-Yau Differential Harnack inequalities for the linear heat equation on a static Riemannian manifold. We will also discuss Hamilton’s matrix inequality on a closed Riemannian manifold and Harnack inequalities for the Ricci flow which is a nonlinear heat equation for time dependent Riemannian metric. The Ricci flow has been applied to prove some striking topological results such as Thurston’s Geometrization conjecture, Poincare conjecture and Differentiable sphere theorem. Li-Yau Harnack inequalities on the other hand have been used to prove heat kernel bounds, Laplacian eigenvalues bounds and Liouville type theorems. Furthermore, we will discuss Perelman entropy functionals and L-length functional with respect to the Ricci solitons, which are special solutions to the Ricci flow. We will also discuss some results on estimating the fundamental solution to the conjugate heat equation coupled with the Ricci flow and some of its applications.