  For one-dimensional growth processes we consider the distribution of the
height above a given point of the substrate and study its scale invariance in
the limit of large times. We argue that for self-similar growth from a single
seed the universal distribution is the Tracy-Widom distribution from the theory
of random matrices and that for growth from a flat substrate it is some other,
only numerically determined distribution. In particular, for the polynuclear
growth model in the droplet geometry the height maps onto the longest
increasing subsequence of a random permutation, from which the height
distribution is identified as the Tracy-Widom distribution.
