Two-dimensional crack problems of homogeneous, anisotropic, linear elasticity are solved using the Riemann–Hilbert method. To this end, the Riemann–Hilbert problem of line-discontinuity is formulated for anisotropic plane problems and the necessary parameters and functions are identified. For illustration, the method is applied to obtain the complete stress field and the stress intensity factors for a crack in an infinite anisotropic plate which is loaded on a part of one of its faces. Then, the well-established method of continuously distributed edge-dislocations is considered and illustrated via some example problems; e.g., an infinite anisotropic plate under uniform farfield loads containing: 1. a closed frictional crack and a pair of arbitrarily-located single edge-dislocations, and 2. an infinite row of equally-spaced parallel open cracks. The illustrative examples reveal that the first method offers an effective solution technique for problems where unbalanced tractions are applied on crack surfaces, whereas for problems with self-equilibrating loads applied on the crack faces, the second method is generally well suited. In addition, the method of resultant forces along the crack is discussed and its formulation in terms of the dislocation density functions and also the crack-opening displacements (which is new) is presented. The solutions to some of the example problems are provided in some detail, and for others, just the key formulae (e.g., stress functions and stress intensity factors) are calculated and analyzed. In brief, this paper presents the generalization of the Riemann–Hilbert method from isotropic to anisotropic in-plane elasticity problems, and also provides a collection of certain basic two-dimensional anisotropic crack problems; some of the results here are also new.