  The Kardar-Parisi-Zhang (KPZ) universality class describes a broad range of
non-equilibrium fluctuations, including those of growing interfaces, directed
polymers and particle transport, to name but a few. Since the year 2000, our
understanding of the one-dimensional KPZ class has been completely renewed by
mathematical physics approaches based on exact solutions. Mathematical physics
has played a central role since then, leading to a myriad of new developments,
but their implications are clearly not limited to mathematics -- as a matter of
fact, it can also be studied experimentally. The aim of these lecture notes is
to provide an introduction to the field that is accessible to non-specialists,
reviewing basic properties of the KPZ class and highlighting main physical
outcomes of mathematical developments since the year 2000. It is written in a
brief and self-contained manner, with emphasis put on physical intuitions and
implications, while only a small (and mostly not the latest) fraction of
mathematical developments could be covered. Liquid-crystal experiments by the
author and coworkers are also reviewed.
