  Let $(Z_n)_{n\geqslant 0}$ be a branching process in a random environment
defined by a Markov chain $(X_n)_{n\geqslant 0}$ with values in a finite state
space $\mathbb X$ starting at $X_0=i \in\mathbb X$. We extend from the i.i.d.
environment to the Markovian one the classical classification of the branching
processes into critical and strongly, intermediate and weakly subcritical
states. In all these cases, we study the asymptotic behaviour of the
probability that $Z_n>0$ as $n\to+\infty$.
