  The master equation describing non-equilibrium one-dimensional problems like
diffusion limited reactions or critical dynamics of classical spin systems can
be written as a Schr\"odinger equation in which the wave function is the
probability distribution and the Hamiltonian is that of a quantum chain with
nearest neighbor interactions. Since many one-dimensional quantum chains are
integrable, this opens a new field of applications. At the same time physical
intuition and probabilistic methods bring new insight into the understanding of
the properties of quantum chains. A simple example is the asymmetric diffusion
of several species of particles which leads naturally to Hecke algebras and
$q$-deformed quantum groups. Many other examples are given. Several relevant
technical aspects like critical exponents, correlation functions and
finite-size scaling are also discussed in detail.

High Energy Physics - Theory;Condensed Matter;High Energy Physics - Lattice