Titanium aluminides are technologically important intermetallic alloys with many properties interesting also from a basic research point of view. When alloyed with Mo, several(meta)stable phases have been reported; their properties are, largely unknown due to the alloy processing and/or non-existence as a single-phase material. We employed first principles calculations to study compositional trends in structural and mechanical properties. We could show that Mo increases the density of all studied phases, leads to their chemical destabilization with the exception of the ordered bcc?o phase, increases their ductility, and enhances the elastic anisotropy. Anisotropic thermal expansion coefficients of tetragonal?-TiAl and hexagonal?2Ti3Al phases were calculated. The predicted values show that for?-TiAl, the more computational demanding method with decoupled impact of volume and temperature effects on the cell shape leads to significantly better results than that with only ground-state optimized unit cell geometry. Predictions of ordering temperatures solely based on the configurational entropy do not yield values in the experimentally expected ranges. Furthermore, bcc-fcc structural transformations of?/?o??dis/?TiAl+Mo are studied. In particular, tetragonal (Bain�s path) and trigonal transformations are combined with the concept of special quasi-random structures and examined. Our calculations of the ordered phases show that the?o??tetragonaltransformation is barrierless, i.e., proceeds spontaneously, reflecting the genuine structural instability of the?o phase. Upon alloying of?7.4 at.% Mo, a small barrier between?oand?-related local energy minima is formed. Yet a higher Mo content of?9 at.% leads to an opposite-direction barrierless transformation???o, i.e., fully stabilizing the?o phase. The martensitic bcc-hcp transformation for TiAl+Mo alloys were reported. Since the Potential Energy Surfaces (PES) clearly suggest that the minimum energy paths are not straight connections of the initial and final states, we have additionally relaxed the ionic positions along the transformation paths. We could show that elastic energy is a decent approximation of PES as a function of cell shape and fixed atomic positions, provided the initial structure is mechanically stable. The transformation energy landscapes a function of Mo content predicts that, adding Mo favors ?o/? phase on the expense of B19/??, eventually leading for spontaneous, barrierless transformations B19??o and ???? for 12.5 at.% Mo.