The incompatibility operator arises in the modeling of elastic materials with dislocations and in the intrinsic approach to elasticity, where it is related to the Riemannian curvature of the elastic metric. It consists of applying successively the curl to the rows and the columns of a second-rank tensor, usually chosen symmetric and divergence-free. This paper presents a systematic analysis of boundary value problems associated with the incompatibility operator. It provides answers to such questions as existence and uniqueness of solutions, boundary trace lifting, and transmission conditions. Physical interpretations in dislocation models are also discussed, but the application range of these results far exceed any specific physical model.