The Khardar-Parisi-Zhang (KPZ) equation is a stochastic
partial differential equation used to describe randomly evolving
interfaces. Its solution has an unusual sclaing behaviour, and the
distribution of the fluctuations are related to random matrices. The
class of such models is called the KPZ universality class and is
predicted to contain a number of discrete and semi-discrete models.
We will discuss some of these models and recent progress made towards
establishing this universality. Examples include polymer models and
interacting particle systems.