  We obtain exact travelling wave solutions for three families of stochastic
one-dimensional nonequilibrium lattice models with open boundaries. These
solutions describe the diffusive motion and microscopic structure of (i) of
shocks in the partially asymmetric exclusion process with open boundaries, (ii)
of a lattice Fisher wave in a reaction-diffusion system, and (iii) of a domain
wall in non-equilibrium Glauber-Kawasaki dynamics with magnetization current.
For each of these systems we define a microscopic shock position and calculate
the exact hopping rates of the travelling wave in terms of the transition rates
of the microscopic model. In the steady state a reversal of the bias of the
travelling wave marks a first-order non-equilibrium phase transition, analogous
to the Zel'dovich theory of kinetics of first-order transitions. The stationary
distributions of the exclusion process with $n$ shocks can be described in
terms of $n$-dimensional representations of matrix product states.
