In this thesis, we study the possibility of applying discontinuous Galerkin (DG) methods to elliptic interface problems. This problem arises, for example, in two-phase flow simulations when the projection method is used to solve for the pressure. The main characteristic of interface problems is that the solutions are discontinuous and so are their derivatives. The discontinuities are in fact prescribed on an interface (a co-dimensional manifold). Here, we assume we have a triangulation of the computational domain that perfectly fits this interface. A standard way to solve this problem with finite element methods is to enforce the discontinuity of the solution in the finite element space. The method presented here differs from such methods in that all conditions (Dirichlet and jump conditions) are implemented weakly. We show that the method is optimally convergent in the $L^2$ -norm, and check this result by numerical experiments. Finally, we apply our method to a problem with dynamic boundary conditions. This problem arises as a significant part of the study of the electroporation phenomenon which has important applications to gene therapy and cancer treatment.