Survival of branching processes in random environments

This review paper presents the known results on the asymptotics of the survival probability and limit theorems conditioned on survival of critical and subcritical branching processes in IID random environments. The key assumptions of the family of population models in question are: non-overlapping generations, independent reproduction of particles within a generation, independent reproduction laws between generations. This is a biologically important generalization of the time inhomogeneous branching processes. The assumption of IID (independent and identically distributed) random environments reflects uncertainty in the future (as well as historical) reproduction regimes in actual populations. This review focusses on a particular range of questions of prime interest for the authors. The reader should be aware of the fact that there are many very interesting papers covering other issues on branching processes in varying and random environments which are not mentioned here.