We review recent progress in the loss or transfer of the polarisation of angular momentum in collisions of small free radicals with a thermal bath of closed-shell partners. Our primary theme is the connection between the observed behaviour and the nature of the underlying potential energy surfaces (PESs). We consider the systems NO(A(2)Sigma(+)) + He and Ar; OH(A(2)Sigma(+)) + He and Ar; OH(X-2 Pi) + He, Ar and Xe; NO(X-2 Pi) + Ar and CN(AX(2)Pi) + Ar, which vary both kinematically and in the strength of any attractive interaction. They are chosen because reliable theoretical PESs have been predicted; QS or QCT scattering calculations have been carried out; and they have been studied by recently developed experimental methods sensitive to polarisation. The efficiency of elastic depolarisation depends intimately on the competition with other, inelastic outcomes. It is generally found to be inefficient for systems dominated by impulsive forces, which promote instead changes of state. This is exacerbated by kinematic effects, in particular for the light He collision partner. Moderately attractive PESs support more efficient elastic depolarisation for low rotational levels, but this drops off rapidly with increasing rotation. Persistent elastic depolarisation across all rotational levels is a feature of deeply attractive, strongly anisotropic PESs. The (2)Sigma(+)-rare gas systems are characterised by a single controlling PES and are well described by a spin-spectator model. The (2)Pi-rare gas systems require two PESs, expressible as (2)A' and (2)A '' adiabatic surfaces or their diabatic sum and difference, V-sum and V-diff. Propensities rules reflecting the symmetries of these surfaces help to explain much of the detailed behaviour.