Abstract In the interstellar medium, inelastic collisions are so rare that they cannot maintain a local thermodynamical equilibrium (LTE). Atomic and molecular populations therefore do not follow a simple Boltzmann distribution and non-LTE spectra are the rule rather than the exception. In such conditions, accurate state-to-state collisional data are crucial for a quantitative interpretation of spectra. In recent years, considerable progress has been made in quantum calculations of inelastic cross sections for a variety of targets, types of transitions and projectiles. For a few benchmark species, detailed comparisons between theory and experiment were also carried out at the state-to-state level and in the quantum regime. In this article, we highlight such comparisons for three important molecules: CO, H 2 O and CH + . We also describe current computational efforts to extend these advances to ever larger targets, new transition types, and new environments (e.g. stellar envelopes or cometary atmospheres).