The Kardar-Parisi-Zhang (KPZ) equation is a nonlinear stochastic
differential equation which describes surface growth.
We consider the one-dimensional version of the equation with
sharp wedge initial conditions. We show that the distributions
of the height is written as an integral of a Fredholm determinant.

We discuss a few properties of the solution. In the long time
limit it tends to the GUE Tracy-Widom distribution. The first order
correction is of t^{-1/3} which is consistent with a recent
experiment of liquid crystal turbulence. We also explain the
derivation of our results based on the contour integral formula for
ASEP by Tracy and Widom.

This is based on a collaboration with H. Spohn.