The large scale behaviour of microscopic stochastic particle systems can often be described in tems of nonlinear partial differential equations which can be predicted phenomenologically or sometimes derived rigorously using probabilistic tools. For one-component systems this allows not only for computing the (deterministic) space-time evolution of the coarse-grained local order parameter, but also for the derivation of the stationary phase diagram in bulk-driven finite systems with open boundaries. The Bethe ansatz provides the means to study fluctuations on finer scales. Systems with two or more components exhibit richer physics, but the theory is far less developed, both mathematically from a probabilistic and PDE point of view and from a statistical physics perspective. Focussing on paradigmatic one-dimensional lattice gas models for driven diffusive systems far from thermal equilibrium the lecture aims at giving a non-technical overview of some well-known rigorous and some more recent numerically established results for one-component systems with conserved particle dynamics or with slow reaction kinetics and at highlighting some aspects of the present incomplete state of art for two-component systems which deserve further investigation.