  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.
