Abstract: In 1870 Jordan explained how Galois theory can be applied
to problems from enumerative geometry, with the group encoding
intrinsic structure of the problem. Earlier Hermite showed
the equivalence of Galois groups with geometric monodromy
groups, and in 1979 Harris initiated the modern study of
Galois groups of enumerative problems. He posited that
a Galois group should be `as large as possible' in that it
will be the largest group preserving internal symmetry in
the geometric problem.
I will describe this background and discuss some work
in a long-term project to compute, study, and use Galois
groups of geometric problems, including those that arise
in applications of algebraic geometry.