  We develop a criterion for transience for a general model of branching Markov
chains. In the case of multi-dimensional branching random walk in random
environment (BRWRE) this criterion becomes explicit. In particular, we show
that \emph{Condition L} of Comets and Popov is necessary and sufficient for
transience as conjectured. Furthermore, the criterion applies to two important
classes of branching random walks and implies that the critical branching
random walk is transient resp. dies out locally.
