Summary: We present a high-order hybridizable discontinuous Galerkin method for solving elliptic interface problems in which the solution and gradient are nonsmooth because of jump conditions across the interface. The hybridizable discontinuous Galerkin method is endowed with several distinct characteristics. First, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby leading to a significant reduction in the global degrees of freedom. Second, they provide, for elliptic problems with polygonal interfaces, approximations of all the variables that converge with the optimal order of \(k+1\) in the \(L^{2}(\Omega)\)-norm where \(k\) denotes the polynomial order of the approximation spaces. Third, they possess some superconvergence properties that allow the use of an inexpensive element-by-element postprocessing to compute a new approximate solution that converges with order \(k+2\). However, for elliptic problems with finite jumps in the solution across the curvilinear interface, the approximate solution and gradient do not converge optimally if the elements at the interface are isoparametric. The discrepancy between the exact geometry and the approximate triangulation near the curved interfaces results in lower order convergence. To recover the optimal convergence for the approximate solution and gradient, we propose to use superparametric elements at the interface.