The core structure of superdislocations in L10 TiAl was investigated with a view to clarifying their dissociation abilities and the mechanisms by which they may become sessile by self-locking. A detailed knowledge of the fine structure of dislocations is essential in analysing the origin of the various deformation features. Atomistic simulation of the core structure and glide of the screw superdislocation was carried out using a bond order potential for γ-TiAl. The core structure of the screw superdislocation was examined, starting with initial unrelaxed configurations corresponding to various dislocation dissociations discussed in the literature. The superdislocation was found to possess in the screw orientation either planar (glissile) or non-planar (sessile) core structures. The response of the core configurations to externally applied shear stress was studied. Some implications were considered of the dissociated configurations and their response to externally applied stress on dislocation dynamics, including the issue of dislocation decomposition, the mechanism of locking and the orientation dependence of the dislocation substructure observed in single-phase γ-TiAl. An unexpectedly rich and complex set of candidate core structures, both planar and non-planar, was found, the cores of which may transform under applied stress with consequent violation of Schmid's law.