  The inverse Galois problem asks whether any finite group can be realised as
the Galois group of a Galois extension of the rationals. This problem and its
refinements have stimulated a large amount of research in number theory and
algebraic geometry in the past century, ranging from Noether's problem (letting
X denote the quotient of the affine space by a finite group acting linearly,
when is X rational?) to the rigidity method (if X is not rational, does it at
least contain interesting rational curves?) and to the arithmetic of
unirational varieties (if all else fails, does X at least contain interesting
rational points?). The goal of the present notes, which formed the basis for
three lectures given at the Park City Mathematics Institute in August 2022, is
to provide an introduction to these topics.
