Analysis on graphs has recently been shown to lead to powerful algorithms in learning, in particular for regression, classification andclustering. Eigenfunctions of the Laplacian on a graph are a natural basis for analyzing functions on a graph, as we have seen in presentations of recent work by partecipants to this conference. In this talk we introduce a new flexible set of basis functions, called Diffusion Wavelets, that allow for a multiscale analysis of functions on a graph, very much in the same way classical wavelets perform a multiscale analysis in Euclidean spaces. They allow efficient, representation, compression, denoising of functions on the graph, and are very well-suited for learning, as well as unsupervised algorithms. They are also associated with a multiscale decomposition of the graph, which has applications by itself. We will discuss this construction with several examples, going from signal processing on manifolds and graphs, to some recent preliminary applications to clustering and learning.