The intermetallic compound TiAl possesses a unique complex of properties that include sufficiently low material density, high values of the strength-to-ductility ratio, high elastic moduli, high oxidation resistance, low creep rate, and improved fatigue characteristics. These properties make TiAl alloys very attractive, particularly for structural applications for aerospace and aeronautic industries, where, at certain temperatures, they might be capable of replacing heavy nickel-based superalloys. However, so far applications of TiAl alloys have been limited by their poor ductility. Many of the recent studies have focused on the source of this limited ductility and on methods to improve this property. It has been found out experimentally that the strength and ductility of gamma-TiAl alloys can be affected by many different parameters, including alloy stoichiometry, heat treatment, deformation temperature, impurity content, grain size, and ternary element additions. In this thesis we present the results of our computer simulations of deformation and fracture in TiAl. In contrast to many previous studies our simulations include the interaction of the crack with point defects in the lattice. We use the molecular statics technique with atomic interactions described in terms of the embedded atom method. We simulate the crack propagation along (100), (001), (110) and (111) planes in TiAl. The cleavage along (100) and (001) planes shows purely brittle behavior, whereas the cleavage along (110) and (111) planes is accompanied by extensive dislocation emission. Our studies of the crack interaction with point defects reveal that vacancies and antisites near the crack tip can influence the amount of plastic deformation. Another important observation is that the antisite formation energy near the crack tip is generally lower than in the perfect lattice. This observation suggests the formation of relatively disordered zones near the crack tip at high temperatures, and leads us to a formulation of a new mechanism of a brittle-to-ductile transition in TiAl.