The Processor Sharing (PS) queue was originally proposed to model time sharing in computer systems. Nowadays it is one of the main tools to descibe congestion of TCP traffic at flow level. From a probabilistic point of view, the PS queue is interesting in view of its connection with Crump-Mode-Jagers Branching processes. In the first part of this talk, I will give a self-contained introduction to the PS queue, and review some recent results for heavy-tailed service times. The second part of this talk is based on current work with Michel Mandjes (CWI). For the GI/GI/1 PS queue, we consider the logarithmic tail asymptotics of the steady-state sojourn-time distribution under light-tailed assumptions. Our proof involves a standard change-of-measure argument and a recent result on overloaded PS queues by Puha, Stolyar and Williams. We apply our result to construct an importance sampling algorithm which is asymptotically optimal.