  Universality is a powerful concept, which enables making qualitative and
quantitative predictions in systems with extensively many degrees of freedom.
It finds realizations in almost all branches of physics, including in the realm
of nonequilibrium systems. Our focus here is on its manifestations within a
specific class of nonequilibrium stationary states: driven open quantum matter.
Progress in this field is fueled by a number of uprising platforms ranging from
light-driven quantum materials over synthetic quantum systems like cold atomic
gases to the functional devices of the noisy intermediate scale quantum era.
These systems share in common that, on the microscopic scale, they obey the
laws of quantum mechanics, while detailed balance underlying thermodynamic
equilibrium is broken due to the simultaneous presence of Hamiltonian unitary
dynamics and nonunitary drive and dissipation. The challenge is then to connect
this microscopic physics to macroscopic observables, and to identify universal
collective phenomena that uniquely witness the breaking of equilibrium
conditions, thus having no equilibrium counterparts. In the framework of a
Lindblad-Keldysh field theory, we discuss on the one hand the principles
delimiting thermodynamic equilibrium from driven open stationary states, and on
the other hand show how unifying concepts such as symmetries, the purity of
states, and scaling arguments are implemented. We then present instances of
universal behavior structured into three classes: new realizations of
paradigmatic nonequilibrium phenomena, including a survey of first experimental
realizations; novel instances of nonequilibrium universality found in these
systems made of quantum ingredients; and genuinely quantum phenomena out of
equilibrium, including in fermionic systems. We also discuss perspectives for
future research on driven open quantum matter.
