Network science is a rapidly growing interdisciplinary area at the intersection of mathematics, physics, computer science, and a multitude of disciplines ranging from the life sciences to the social sciences and even the humanities. Network analysis methods are now widely used in proteomics, the study of social networks (both human and animal), finance, ecology, bibliometric studies, archeology, the evolution of cities, and a host of other fields. After giving a broad overview of network science, I will introduce the audience to some of the more fundamental mathematical and computational problems arising in the analysis of networks, with an emphasis on the basic notions of centrality, communicability, and robustness. I will show how these lead to large-scale sparse numerical linear algebra computations including the solution of linear systems and eigenvalue problems, and the evaluation of functions of matrices. The talk is intended to be accessible to a broad audience.