Topology and symmetry have emerged as compelling guiding principles to predict and harness the propagation of waves in natural and artificial materials. Be it for quantum particles (such as electrons) or classical waves (such as light, sound or mechanical motion), these concepts have so far been mostly developed in idealized scenarios, in which the wave amplitude is neither attenuated nor amplified, and time evolution is unitary. In recent years, however, there has been a considerable push to explore the consequences of topology and symmetries in non-conservative, non-equilibrium or non-Hermitian systems. A plethora of driven artificial materials has been reported, blurring the lines between a wide variety of fields in physics and engineering, including condensed matter, photonics, phononics, optomechanics, as well as electromagnetic and mechanical metamaterials. Here we discuss the latest advances, emerging opportunities and open challenges for combining these exciting research endeavours into the new pluridisciplinary field of non-Hermitian topological systems.