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A subfactor is an inclusion of von Neumann algebras with trivial centers. The representation category of a subfactor is a unitary 2-category which generalizes the representation categories of quantum groups. Thus we think of a subfactor as encoding quantum symmetries. Subfactors are connected to mathematical physics in many ways. Most famously, Jones found his knot polynomial via the Temperley-Lieb algebras, which arise in statistical mechanics. When a sub factor is finite depth, its representation category produces a Morita equivalent pair of unitary fusion categories, which yield Turaev-Viro TQFTs. Subfactors also arise naturally from conformal nets in algebraic quantum field theory, as inclusions of local algebras associated to regions of space-time. The quantum double construction produces unitary modular categories, which are important to research in topological phases of matter and topological quantum computation. I'll begin with an introduction to subfactors and fusion categories, and I'll give an overview of some of these applications. Toward the end, I'll also talk a bit about my research on classifying subfactors and fusion categories.