Abstract: Partial differential equations posed on surfaces arise in mathematical models for many natural phenomena: diffusion along grain boundaries, lipid interactions in biomembranes, pattern formation, and transport of surfactants on fluidic interfaces to mention a few. Numerical methods for solving PDEs posed on manifolds recently received considerable attention. In this talk, we discuss finite element methods to solve PDES on both stationary surfaces and surface with prescribed evolution. The focus of the talk is on geometrically unfitted methods, i.e. methods that avoid parametrization and triangulation of surfaces in a common sense. We explain how unfitted discretizations facilitate the development of a fully Eulerian numerical framework and enable easy handling of time-dependent surfaces including the case of topological transitions. We consider two methods falling in this category: a method based on PDE extensions off the surface and its `dual’ that uses the restrictions of outer finite element spaces to solve PDE on the surface. The application of the latter technique known as Trace FEM or Cut FEM is further demonstrated for a sequence of problems of increasing complexity, ranging from the Laplace-Beltrami equation on a fixed domain to the evolving surface Cahn-Hilliard equation, which models lateral phase separation in plasma membranes undergoing deformation and fusion.