The existence of recurrent stochastic processes with stationary independent increments in an arbitrary locally compact group has remained a challenging research topic over a period of four decades. On the other hand, there exists an elaborate theory of transient processes in groups leading to interesting problems in potential theory. In the present expository talk, which is designed to be more or less self contained, the speaker will revisit the well-known Kesten-Spitzer transience criterion for processes in {\it abelian} locally compact groups, and will describe a purely measure theoretic approach to it.