We present higher-order piecewise continuous finite element methods for solving a class of interface problems where the finite element mesh does not fit the interface. The method is based on correction terms added only to the right-hand side in the standard variational formulation of the problem. We prove optimal error estimates of the methods on general quasi-uniform and shape regular meshes in maximum norms. We apply the method to a Stokes interface problem, adding correction terms for the velocity and the pressure, obtaining optimal convergence results. We finally discuss the extension of this technique to transmission problem with high-contrast coefficients. This is a joint work with Johnny Guzman and Manuel Sanchez-Uribe.