We study a discrete time Markov process with particles being able to perform
discrete time random walks and create new particles, known as Branching
Random Walk (BRW). We suppose that there are particles of different types,
and the transition probabilities, as well as offspring distribution, depend
on the type and the position of the particle. Criteria of (strong) recurrence
and transience are presented, and some applications (spatially homogeneous
case, Lamperti BRW, many-dimensional BRW) are studied.