The α/β interface is central to the microstructure and mechanical properties of titanium alloys. We investigate the structure, thermodynamics and migration of the coherent and semicoherent Ti α/β interfaces as a function of temperature and misfit strain via molecular dynamics (MD) simulations, thermodynamic integration and an accurate, DFT-trained Deep Potential. The structure of an equilibrium semicoherent interface consists of an array of steps, an array of misfit dislocations, and coherent terraces. Analysis determines the dislocation and step (disconnection) array structure and habit plane. The MD simulations show the detailed interface morphology dictated by intersecting disconnection arrays. The steps are shown to facilitate α/β interface migration, while the misfit dislocations lead to interface drag; the drag mechanism is different depending on the direction of interface migration. These results are used to predict the nature of α phase nucleation on cooling through the α-β phase transition.