This course is concerned with spectral gaps in graphs, groups, manifolds, and operator algebras, with an emphasis on optimal spectral gaps. Both deterministic and random models will be considered. Topics include Ramanujan graphs, optimal spectral gaps of random graphs and surfaces, spectral gaps of Schreier and Cayley graphs of finite groups, and strong convergence. We cover a variety of tools coming from several areas of mathematics, recent developments, and numerous open problems.