||In this article an unfitted Discontinuous Galerkin Method is proposed to discretize elliptic interface problems. The method is based on the Symmetric Interior Penalty Discontinuous Galerkin Method and can also be interpreted as a generalization of the method given in [A. Hansbo, P. Hansbo, An unfitted finite element method based on Nitsche’s method for elliptic interface problems, Comp. Meth. Appl. Mech. Eng., Vol. 191, (2002), 5537–5552]. We prove the optimal h-convergence of the method for arbitrary p in energy- and in L2 -norm. In fact we present an hp-error estimate. The analysis includes grids with hanging nodes and the proposed DG Method is symmetric and inherits the attractive locality property of general Discontinuous Galerkin methods. A variant of the method is proposed which additionally behaves well with respect to the pointwise error in the gradient. The behaviour of the methods in numerical experiments is displayed.