The Heat Equation arises in a number of different situations and in this talk we will present a version of this equation for graphs (either directed or undirected graphs). The generality of graphs allow one to model many real world problems and attempt to find optimal solutions to these problems. We will show how to solve this version of the Heat Equation on graphs under more and more general assumptions such as non-constant edge weightings and a noisy version of the Heat Equation. This talk should be accessible to all undergraduate math majors.