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The inverse Galois problem asks whether any finite group can be realised asthe Galois group of a Galois extension of the rationals. This problem and itsrefinements have stimulated a large amount of research in number theory andalgebraic geometry in the past century, ranging from Noether's problem (lettingX 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 atleast contain interesting rational curves?) and to the arithmetic ofunirational varieties (if all else fails, does X at least contain interestingrational points?). The goal of the present notes, which formed the basis forthree lectures given at the Park City Mathematics Institute in August 2022, isto provide an introduction to these topics.

Park City lecture notes: around the inverse Galois problem