This is a pedagogical account on reaction-diffusion systems and their
relationship with integrable quantum spin chains.
Reaction-diffusion systems are paradigmatic examples of non-equilibrium
systems. Their long-time behaviour is strongly influenced through fluctuation
effects in low dimensions which renders the habitual mean-field cinetic
equations inapplicable. Starting from the master equation rewritten as a
Schr\"odinger equation with imaginary time, the associated quantum hamiltonian
of certain one-dimensional reaction-diffusion models is closely related to
integrable magnetic chains. The relationship with the Hecke algebra and its
quotients allows to identify integrable reaction-diffusion models and, through
the Baxterization procedure, relate them to the solutions of Yang-Baxter
equations which can be solved via the Bethe ansatz. Methods such as spectral
and partial integrability, free fermions, similarity transformations or
diffusion algebras are reviewed, with several concrete examples treated
explicitly. An outlook on how the recently-introduced concept of local scale
invariance might become useful in the description of non-equilibrium ageing
phenomena is presented, with particular emphasis on the kinetic Ising model
with Glauber dynamics.