In this talk, we present unfitted finite element methods for the numerical solution of the Cahn-Hilliard equation on embedded (or potentially moving) domains. Our approach combines the continuous interior penalty method from [1] with the unfitted finite element techniques introduced in [2,3] to devise geometrically robust schemes for the primal 4th-order Cahn-Hilliard formulation. As a consequence, we are able to derive stability, a priori error, and condition error estimates for the linearized biharmonic equation with Cahn-Hilliard type boundary conditions resembling the classical results from [1] obtained on conform meshes. Afterwards, we demonstrate how the resulting spatial discretization scheme can be combined with BDF-based time-stepping methods to numerically solve the complete Cahn-Hilliard equation on domains which are embedded into a structured background mesh. Finally, we provide ample numerical results to corroborate our theoretical findings and discuss possible advantages and disadvantages of the presented formulations. [1] S. C. Brenner, S. Gu, T. Gudi, and L. Sung. A quadratic C0 interior penalty method for linear fourth order boundary value problems with boundary conditions of the Cahn–Hilliard type. Int. J. Numer. Meth. Engng., Vol. 50(4), pp. 2088–2110, 2012 [2] E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. CutFEM: discretizing geometry and partial differential equations. SIAM J. Numer. Anal., Vol. 104, pp. 472–501, 2015. [3] S. Badia, F. Verdugo, and A.F. Mart ́ın, Alberto F. The Aggregated Unfitted Finite Element Method for Elliptic Problems. Comput. Methods Appl. Mech. Engrg., Vol. 336, pp. 533–553, 2018