  Brownian motion is a continuum scaling limit for a wide class of random
processes, and there has been great success in developing a theory for its
properties (such as distribution functions or regularity) and expanding the
breadth of its universality class. Over the past twenty five years a new
universality class has emerged to describe a host of important physical and
probabilistic models (including one dimensional interface growth processes,
interacting particle systems and polymers in random environments) which display
characteristic, though unusual, scalings and new statistics. This class is
called the Kardar-Parisi-Zhang (KPZ) universality class and underlying it is,
again, a continuum object -- a non-linear stochastic partial differential
equation -- known as the KPZ equation. The purpose of this survey is to explain
the context for, as well as the content of a number of mathematical
breakthroughs which have culminated in the derivation of the exact formula for
the distribution function of the KPZ equation started with {\it narrow wedge}
initial data. In particular we emphasize three topics: (1) The approximation of
the KPZ equation through the weakly asymmetric simple exclusion process; (2)
The derivation of the exact one-point distribution of the solution to the KPZ
equation with narrow wedge initial data; (3) Connections with directed polymers
in random media. As the purpose of this article is to survey and review, we
make precise statements but provide only heuristic arguments with indications
of the technical complexities necessary to make such arguments mathematically
rigorous.
