  These notes are based on a talk given at the 2018 Arizona School of Analysis
and Mathematical Physics. We give a comprehensive introduction to the KPZ
universality class, a conjectured class of stochastic process with local
interactions related to random growth processes in $1+1$ dimensions. We
describe some of the characteristic properties of the KPZ universality class
such as scaling exponents and limiting statistics. In particular, we aim to
extract the characteristic properties of the KPZ universality class by
understanding the KPZ stochastic partial differential equation by a special
discrete approximation given by the asymmetric simple exclusion process (ASEP).
The connection with the ASEP is very important as the process enjoys a rich
integrability structure that leads to many exact formulas.
