Many hyperbolic conservation laws are subject to "wetting and drying", where the system's mass variable becomes effectively zero in portions of the domain and thus introducing "dry" regions. Wetting and drying can cause numerical instability and loss of positivity and accuracy in many numerical methods and is a perennial problem in simulating the shallow water equations for flood modeling and other positivity constrained hyperbolic conservation laws. Here, we explore using moving cuts in finite volume (FV) and discontinuous Galerkin (DG) formulations to capture the wet-dry inferface. Our approach seeks to exploit FV and DG method's inherent tolerance of discontinuities at element boundaries to handle the often numerically singular wet-dry interface. We present our recent work on these cut wetting and drying FV/DG methods and their stability and accuracy.