The extended finite element method (XFEM) enables the accurate approximation of solutions with jumps or kinks within elements. Optimal convergence rates have frequently been achieved for linear elements and piecewise planar interfaces. Higher-order convergence for arbitrary curved interfaces relies on two major issues: (i) an accurate quadrature of the Galerkin weak form for the cut elements and (ii) a careful formulation of the enrichment, which should preclude any problems in the blending elements. For (i), we employ a strategy of subdividing the elements into subcells with only one curved side. Reference elements that are higher-order on only one side are then used to map the integration points to the real element. For (ii), we find that enrichments for strong discontinuities are easily extended to higher-order accuracy. In contrast, problems in blending elements may hinder optimal convergence for weak discontinuities. Different formulations are investigated, including the corrected XFEM. Numerical results for several test cases involving strong or weak curved discontinuities are presented. Quadratic and cubic approximations are investigated. Optimal convergence rates are achieved using the standard XFEM for the case of a strong discontinuity. Close-to-optimal convergence rates for the case of a weak discontinuity are achieved using the corrected XFEM. Copyright © 2009 John Wiley & Sons, Ltd.