  The question of recurrence and transience of branching Markov chains is more
subtle than for ordinary Markov chains; they can be classified in transience,
weak recurrence, and strong recurrence. We review criteria for transience and
weak recurrence and give several new conditions for weak recurrence and strong
recurrence. These conditions make a unified treatment of known and new examples
possible and provide enough information to distinguish between weak and strong
recurrence. This represents a step towards a general classification of
branching Markov chains. In particular, we show that in \emph{homogeneous}
cases weak recurrence and strong recurrence coincide. Furthermore, we discuss
the generalization of positive and null recurrence to branching Markov chains
and show that branching random walks on $\Z$ are either transient or positive
recurrent.

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