The phenomenon of KPZ (Kardar-Parisi-Zhang) universality was a very active research area in the last 10 years. One can say that it describes universal statistical properties of long directed optimal paths (geodesics) in 2D disordered environment. Apart from many physical applications, it is remarkable for extremely strong universality properties in very general setting, as well as for many features of exact integrability.
In this talk we'll give a general overview of the KPZ problem. We shall then discuss a possible approach to the problem of universality based on global solutions to the random Hamilton-Jacobi equation. We shall also discuss a new renormalisation scheme based on geometrical properties of minimisers and shocks.
Finally, we shall formulate several conjectures and discuss some rigorous results in their support.