This presentation focuses on the development and application of Extended Discontinuous Galerkin (XDG) methods which has been originally developed to simulate multiphase flows with high accuracy. Such flows are characterized by the presence of fluid interfaces that separate immiscible fluids. Due to abrupt changes in density and viscosity and due to surface tension effects, there are discontinuities in pressure velocity gradient. The XDG method addresses these challenges by embedding the respective interfaces – or boundaries – in a Cartesian background mesh and introducing cut-cells and respective shape functions that conform to the interface geometry. Thus, the method is capable of resolving discontinuities and (certain classes of) singularities in the solution fields with a high order of accuracy. While originally developed for multiphase flows, the XDG method can be applied to a wide range of problems involving deforming domains and free boundaries, as well as flows with internal interfaces. This includes fluid-structure interaction problems, complex, moving geometries, and even shock waves in compressible flows. This talk aims to provide a comprehensive overview of the XDG methodology. This includes the mathematical formulation of the XDG method, numerical stabilization techniques and solution strategies. Furthermore, various results of the application of the XDG method in multiphase flows will be shown, e.g., droplet dynamics, including wetting- and dewetting and evaporation.