The emerging field of signal processing on graphs combines algebraic and spectral graph theory with computational harmonic analysis to process high-dimensional data that naturally resides on the vertices of weighted graphs. This tutorial overview discusses the main challenges in this area, including the irregular structure of graph data domains and the need to incorporate graph structure into localized transform methods. It reviews different ways to define graph spectral domains, which are analogues to the classical frequency domain, and highlights the importance of incorporating the irregular structures of graph data domains. The paper also surveys methods for generalizing fundamental operations such as filtering, translation, modulation, dilation, and downsampling to the graph setting, and reviews localized, multiscale transforms that efficiently extract information from high-dimensional data on graphs. Finally, it concludes with a discussion of open issues and possible extensions.The emerging field of signal processing on graphs combines algebraic and spectral graph theory with computational harmonic analysis to process high-dimensional data that naturally resides on the vertices of weighted graphs. This tutorial overview discusses the main challenges in this area, including the irregular structure of graph data domains and the need to incorporate graph structure into localized transform methods. It reviews different ways to define graph spectral domains, which are analogues to the classical frequency domain, and highlights the importance of incorporating the irregular structures of graph data domains. The paper also surveys methods for generalizing fundamental operations such as filtering, translation, modulation, dilation, and downsampling to the graph setting, and reviews localized, multiscale transforms that efficiently extract information from high-dimensional data on graphs. Finally, it concludes with a discussion of open issues and possible extensions.