Integral equation methods have become increasingly practical and powerful for a wide range of problems with interfaces, including fluid dynamics, wave scattering, and electromagnetics. However, the dense linear systems that arise place a premium on accuracy per degree of freedom, making high-order methods and fast algorithms central components of efficient solvers. Recent developments, including high-order quadrature for near- and weakly-singular integrals, geometry-aware discretizations, fast direct solvers, and fast summation methods, have significantly expanded the range of tractable problems. Such challenges arise in problems with interfaces exhibiting high curvature, periodic structures, or near-contact between distinct components. This minisymposium will bring together researchers across these areas to highlight common challenges and emerging high-order integral equation methods for interface problems.