We review recent results from computer simulation studies of polymer glasses, from the chain dynamics around the glass transition temperature Tg to the mechanical behaviour below Tg. These results clearly show that modern computer simulations are able to address and give clear answers to some important issues in the field, in spite of the obvious limitations in terms of length and time scales. In the present review we discuss the cooling rate effects, and the dynamic slowing down of different relaxation processes when approaching Tg for both model and chemistry-specific polymer glasses. The impact of geometric confinement on the glass transition is discussed in detail. We also show that computer simulations are very useful tools to study structure and mechanical response of glassy polymers. The influence of large deformations on mechanical behaviour of polymer glasses in general, and strain hardening effect in particular are reviewed. Finally, we suggest some directions for future research, which we believe will be soon within the capabilities of state of the art computer simulations, and correspond to problems of fundamental interest.