My research concerns the development, analysis, and implementation of computational methods for solving Partial Differential Equations (PDEs) on evolving domains. Such PDEs occur for example in multiphase flow simulations where the moving domains are interfaces separating immiscible fluids or domains that have these interfaces as boundaries. Finite Element Methods (FEM) are well known for efficiently solving PDEs in complex geometries. However, when the geometry undergoes large deformations standard FEM requires re-meshing and interpolation which is often both expensive and cumbersome especially in three space dimensions. In such cases, we provide an alternative: Cut Finite Element Methods (CutFEM). These methods solve PDEs in evolving geometries that cut through a fixed background mesh in an arbitrary fashion without loosing accuracy and without problems with ill-conditioned linear systems.