  Branching processes $(Z_n)_{n \ge 0}$ in a varying environment generalize the
Galton-Watson process, in that they allow time-dependence of the offspring
distribution. Our main results concern general criteria for a.s. extinction,
square-integrability of the martingale $(Z_n/\mathbf E[Z_n])_{n \ge 0}$,
properties of the martingale limit $W$ and a Yaglom type result stating
convergence to an exponential limit distribution of the suitably normalized
population size $Z_n$, conditioned on the event $Z_n >0$. The theorems
generalize/unify diverse results from the literature and lead to a
classification of the processes.
