This paper presents diffusion filtering as a method to smooth signals defined on the nodes of a graph or network. Diffusion filtering considers the given signals as initial temperature distributions in the nodes and diffuses heat through the edges of the graph. The filtered signal is determined by the accumulated temperatures over time at each node. We show multiple other interpretations of diffusion filtering and describe how it can be generalized to encompass a wide class of networks making it suitable for real-world applications. We prove that diffused signals are stable to perturbations in the underlying network. Further, we demonstrate how diffusion filtering can be applied to improve the performance of recommendation systems by considering the problem of predicting ratings from a signal processing perspective.