  This article is firstly a historic review of the theory of Riemann-Hilbert
problems with particular emphasis placed on their original appearance in the
context of Hilbert's 21st problem and Plemelj's work associated with it. The
secondary purpose of this note is to invite a new generation of mathematicians
to the fascinating world of Riemann-Hilbert techniques and their modern
appearances in nonlinear mathematical physics. We set out to achieve this goal
with six examples, including a new proof of the integro-differential
Painlev\'e-II formula of Amir, Corwin, Quastel \cite{ACQ} that enters in the
description of the KPZ crossover distribution. Parts of this text are based on
the author's plenary lecture at the $15$th International Symposium on
Orthogonal Polynomials, Special Functions and Applications (OPSFA) in
Hagenberg, Austria.
