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.