  Many systems that can be described in terms of diffusion-limited `chemical'
reactions display non-equilibrium continuous transitions separating active from
inactive, absorbing states, where stochastic fluctuations cease entirely. Their
critical properties can be analyzed via a path-integral representation of the
corresponding classical master equation, and the dynamical renormalization
group. An overview over the ensuing universality classes in single-species
processes is given, and generalizations to reactions with multiple particle
species are discussed as well. The generic case is represented by the processes
A <-> A + A, and A -> 0, which map onto Reggeon field theory with the critical
exponents of directed percolation (DP). For branching and annihilating random
walks (BARW) A -> (m+1) A and A + A -> 0, the mean-field rate equation predicts
an active state only. Yet BARW with odd m display a DP transition for d <= 2.
For even offspring number m, the particle number parity is conserved locally.
Below d_c' = 4/3, this leads to the emergence of an inactive phase that is
characterized by the power laws of the pair annihilation process. The critical
exponents at the transition are those of the `parity-conserving' (PC)
universality class. For local processes without memory, competing pair or
triplet annihilation and fission reactions k A -> (k - l) A, k A -> (k+m)A with
k=2,3 appear to yield the only other universality classes not described by
mean-field theory. In these reactions, site occupation number restrictions play
a crucial role.
