We establish some optimal a priori error estimates on some variants of the eXtended Finite Element Method (Xfem), namely the Xfem with a cut-off function and the standard Xfem with a fixed enrichment area. Both the Lamé system (homogeneous isotropic elasticity) and the Laplace problem are considered. The convergence of the numerical stress intensity factors is also investigated. Some numerical experiments corroborating the theoretical results are presented. Copyright © 2011 John Wiley & Sons, Ltd.