Boundary integral equations (BIEs) are a useful framework for reducing partial differential equations to the boundaries and interfaces of a physical domain. Though these methods result in dense matrices and typically require an explicit Green's function, recent advances in the analysis and implementation of these methods have made it easier to handle complex wave problems and incorporate new physics such as hydro-, elasto-, and electrodynamics. This session highlights these advances by bringing together recent work on both mathematical formulations of BIEs and fast computational tools for solving them. Topics include skeletonization and hierarchical compression, broadband solvers, periodic summation techniques, Maxwell waveguides, and ice-ocean interactions.