The inverse Galois problem for a field $K$ asks what finite groups
can arise as Galois groups of extensions of $K$. This classical
problem admits a vast generalization, which would take into account not
only finite extensions of $K$ but also features of the topology of
algebraic varieties defined by polynomials with coefficients in
$K$. In this generalization, not only finite groups but also algebraic
groups arise as the relevant symmetry groups. I will motivate this general
problem and discuss some work on finding the exceptional algebraic
groups out in the arithmetic wild.