  A systematic theory for the diffusion--limited reaction processes $A + A \to
0$ and $A \to (m+1) A$ is developed. Fluctuations are taken into account via
the field--theoretic dynamical renormalization group. For $m$ even the mean
field rate equation, which predicts only an active phase, remains qualitatively
correct near $d_c = 2$ dimensions; but below $d_c' \approx 4/3$ a nontrivial
transition to an inactive phase governed by power law behavior appears. For $m$
odd there is a dynamic phase transition for any $d \leq 2$ which is described
by the directed percolation universality class.
