Many important phenomena in material science, cell biology, and geometry can be formulated as curvature driven interface problems, including phenomena such as reshaping of cell membranes and melting and solidification processes. In this talk, we study cut finite element method [1] (CutFEM)–based discretizations as a flexible computational strategy for evolving interface problems without requiring remeshing. We review the key methodological ingredients of the approach, including level set evolution, reinitialization procedure, and the overall computational workflow for simulations on moving domains. The framework is designed to accommodate a spectrum of interface coupled multiphysics problems. Examples include two phase flow, incompressible flow with embedded rigid body motion, flow coupled to complex interface dynamics, and situations involving topological changes. The approach is implemented on a structured octree based background mesh within the Gridap ecosystem [2] and supports distributed memory parallel execution. [1] E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. CutFEM: discretizing geometry and partial differential equations. Int. J. Numer. Meth. Engng., Vol. 104, pp. 472–501, 2015. [2] S. Badia and F. Verdugo, “Gridap: An extensible Finite Element toolbox in Julia,” J. Open Source Softw., Vol. 5, No. 52, pp. 2520, 2020.