Over the past few decades we have been experiencing an explosion of information generated by large networks of sensors and other data sources. Much of this data is intrinsically structured, such as traffic evolution in a transportation network, temperature values in different geographical locations, information diffusion in social networks, functional activities in the brain, or 3D meshes in computer graphics. The representation, analysis, and compression of such data is a challenging task and requires the development of new tools that can identify and properly exploit the data structure. In this thesis, we formulate the processing and analysis of structured data using the emerging framework of graph signal processing. Graphs are generic data representation forms, suitable for modeling the geometric structure of signals that live on topologically complicated domains. The vertices of the graph represent the discrete data domain, and the edge weights capture the pairwise relationships between the vertices. A graph signal is then defined as a function that assigns a real value to each vertex. Graph signal processing is a useful framework for handling efficiently such data as it takes into consideration both the signal and the graph structure. In this work, we develop new methods and study several important problems related to the representation and structure-aware processing of graph signals in both centralized and distributed settings. We focus in particular in the theory of sparse graph signal representation and its applications and we bring some insights towards better understanding the interplay between graphs and signals on graphs. First, we study a novel yet natural application of the graph signal processing framework for the representation of 3D point cloud sequences. We exploit graph-based transform signal representations for addressing the challenging problem of compression of data that is characterized by dynamic 3D positions and color attributes. Next, we depart from graph-based transform signal representations to design new overcomplete representations, or dictionaries, which are adapted to specific classes of graph signals. In particular, we address the problem of sparse representation of graph signals residing on weighted graphs by learning graph structured dictionaries that incorporate the intrinsic geometric structure of the irregular data domain and are adapted to the characteristics of the signals. Then, we move to the efficient processing of graph signals in distributed scenarios, such as sensor or camera networks, which brings important constraints in terms of communication and computation in realistic settings. In particular, we study the effect of quantization in the distributed processing of graph signals that are represented by graph spectral dictionaries and we show that the impact of the quantization depends on the graph geometry and on the structure of the spectral dictionaries. Finally, we focus on a widely used graph process, the problem of distributed average consensus in a sensor network where sensors exchange quantized information with their neighbors. We propose a novel quantization scheme that depends on the graph topology and exploits the increasing correlation between the values exchanged by the sensors throughout the iterations of the consensus algorithm.