Hyperbolic conservation laws are a class of partial differential equations governing a variety of important applications from coastal flooding to supersonic aerodynamics. Simulation of hyperbolic conservation laws presents a number of challenges, such as long time scales, the ability to produce shocks, and large, geometrically complex, and potentially moving/deforming domains. Discontinuous Galerkin (DG) and finite volume (FV) methods on cut meshes are uniquely equipped to address these challenges provided they can overcome hurdles of their own: entropy stability, the small cell problem, cut cell quadrature, and, in the case of moving boundaries, interface reconstruction/tracking. In this talk I will share my research in cut DG/FV methods capable of overcoming these hurdles and recent work in adapting these methods to tracking the wet-dry front in storm surge simulation.