A methodology for solving three-dimensional crack problems with geometries that are independent of the mesh is described. The method is based on the extended 5nite element method, in which the crack discontinuity is introduced as a Heaviside step function via a partition of unity. In addition, branch functions are introduced for all elements containing the crack front. The branch functions include asymptotic near-tip fields that improve the accuracy of the method. The crack geometry is described by two signed distance functions, which in turn can be de5ned by nodal values. Consequently, no explicit representation of the crack is needed. Examples for three-dimensional elastostatic problems are given and compared to analytic and benchmark solutions. The method is readily extendable to inelastic fracture problems.