  In this minireview we will discuss recent progress in the analytical study of
current-carrying non-equilibrium steady states (NESS) that can be constructed
in terms of a matrix product ansatz. We will focus on one-dimensional exactly
solvable strongly correlated cases, and will study both quantum models, and
classical models which are deterministic in the bulk. The only source of
classical stochasticity in the time-evolution will come from the boundaries of
the system. Physically, these boundaries may be understood as Markovian baths,
which drive the current through the system. The examples studied include the
open XXZ Heisenberg spin chain, the open Hubbard model, and a classical
integrable reversible cellular automaton, namely the Rule 54 of Bobenko {\em et
al}. [Commun. Math. Phys. {\bf 158}, 127 (1993)] with stochastic boundaries.
The quantum NESS can be at least partially understood through the Yang-Baxter
integrability structure of the underlying integrable bulk Hamiltonian, whereas
for the Rule 54 model NESS seems to come from a seemingly unrelated
integrability theory. In both the quantum and the classical case, the
underlying matrix product ansatz defining the NESS also allows for construction
of novel conservation laws of the bulk models themselves. In the classical
case, a modification of the matrix product ansatz also allows for construction
of states beyond the steady state (i.e., some of the decay modes -- Liouvillian
eigenvectors of the model). We hope that this article will help further the
quest to unite different perspectives of integrability of NESS (of both quantum
and classical models) into a single unified framework.
