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The internal twin-boundaries in lamellar γ -TiAl alloys, namely true-twin (TT), rotational boundary (RB), and pseudo-twin (PT), are known to be effective in strengthening the TiAl microstructures. Nevertheless, for designing microstructures with optimised mechanical properties, a better understanding of the role of these boundaries on fracture behavior is still required. To this end, we study how and to what degree crack advancement is affected by the local lattice orientation and atomic structure at the various twin boundaries. Molecular statics simulations were performed in conjunction with a linear elastic fracture mechanics based analysis, to understand the inter-lamellar and as well as trans-lamellar crack advancement at a TT, RB, and PT interface. The fracture toughness as well as the crack advancement mechanisms of the inter-lamellar cracks depend critically on the propagation direction. For instance, cracks along <11-2] in the TT, RB, and PT plane always emit dislocations at the crack tip, while the cracks along the opposite direction are brittle in nature. When it comes to trans-lamellar crack advancement, the crack tip shows significant plastic deformation and toughening for all interfaces. However, at a TT, a brittle crack is able to penetrate through the interface at a higher applied load, and propagates in the adjacent γ' phase, while in the case of RB and PT, the crack tip is blunted and arrested at or near the boundary, resulting in dislocation emission and crack tip toughening. This suggests that a variation of the sequence of the different rotational boundaries could be a possibility to tune the crack tip plasticity and toughening in lamellar TiAl.