The intersection of dynamical systems and networks are used to model a huge variety of phenomena. From social networks, to traffic routes and self-driving cars, to swarms of robots and multiagent systems, to individuals moving about in a geographical area, networks can represent an enormous variety of interacting systems. These networks are typically represented via graphs, which are mathematical objects that encode the information of both the entities and relationships within a network. These graphs in turn can be represented by graph matrices, which encode the relationships between entities in the network. The specific problems that this thesis considers span the learning and control of network phenomena via graph matrices: learning structural correspondences across correlated networks; identification of unknown networked dynamical systems with known control inputs; the generalization performance of machine learning algorithms for graphs and hypergraphs; and machine learning and data-driven control of an epidemic spreading across a network. The key intuition throughout is that many properties of networks and networked dynamical systems can be understood by examining the eigenvalue spectrum of an associated graph matrix. Following this intuition, we propose an algorithm for network alignment using spectral information called SPECTRE that exhibits state-of-the-art performance for aligning networks that are only moderately correlated. Next, we present a method for learning the spectra of a graph matrix using only the sparse output measurements of a networked dynamical system. We further propose a new architecture for signal processing on higher-order graphs, along with a new generalization bound on the performance of graph neural networks via spectral similarity. This generalization result is valid for arbitrary graphs regardless of their structure, engendering the first bound on the generalization performance of a machine learning approach for higher-order graphs. Finally, we present a data-driven framework for multi-task learning and non-linear control of epidemics.