This paper investigates the accuracy of high-order extended ﬁnite element methods (XFEMs) for the solution of discontinuous problems with both straight and curved weak discontinuities in two dimensions. The modiﬁed XFEM, a speciﬁc form of the stable generalised ﬁnite element method (SGFEM), is found to oﬀer advantages in cost and complexity over other approaches, but suﬀers from suboptimal rates of convergence due to spurious higher order contributions to the approximation space. An improved modiﬁed XFEM is presented, with basis functions ‘corrected’ by projecting out higher order contributions that cannot be represented by the standard ﬁnite element basis. The resulting corrections are independent of the equations being solved and need be pre-computed only once for geometric elements of a given order. An accurate numerical integration scheme that correctly integrates functions with curved discontinuities is also presented. Optimal rates of convergence are then recovered for Poisson problems with both straight and quadratically curved discontinuities for approximations up to order 푝≤4. These are the ﬁrst truly optimal convergence results achieved using the XFEM for a curved weak discontinuity, and also the ﬁrst optimally convergent results achieved using the modiﬁed XFEM for any problem with approximations of order 푝 > 1. Almost optimal rates of convergence are recovered for an elastic problem with a circular weak discontinuity for approximations up to order 푝≤4.