Subcritical branching processes in random environment

Let $Z_{n},n=0,1,...$ be a branching process in random environment specified by iid (random) probability generating functions $ f_{0}(s),f_{1}(s),...,f_{n}(s),...$ Such process is called subcritical if $ \mathbf{E}\log f_{0}^{\prime }(1)<0$. Let
\begin{equation*} S_{0}=0,S_{n}=\log f_{0}^{\prime }(1)+\log f_{1}^{\prime }(1)+...+\log f_{n-1}^{\prime }(1),n\geq 1 \end{equation*}
be the associated random walk for such a process. It is known that the set of all subcritical branching processes in random environment may be divided into 4 classes depending on the properties of the distributions of the increments of $S_n.$
We give a survey of the recent results dealing with the survival probabilities of the mentioned classes of subcritical branching processes and with the Yaglom type functional limit theorems for the number of particles in such processes given their survival up to a distant moment.
These results are obtained in collaboration with V.I.Afanasyev (Steklov Mathematical Institute), C.Boeinghoff, G.Kersting, J.Geiger (Frankfurt), and X.Zehng (Hong Kong).