  We give three different criteria for transience of a Branching Markov Chain.
These conditions enable us to give a classification of Branching Random Walks
in Random Environment (BRWRE) on Cayley Graphs in recurrence and transience.
This classification is stated explicitly for BRWRE on $\Z^d.$ Furthermore, we
emphasize the interplay between Branching Markov Chains and the spectral
radius. We prove properties of the spectral radius of the Random Walk in Random
Environment with the help of appropriate Branching Markov Chains.
