Galois Groups via Numerical Algebraic Geometry Galois groups, which are a mainstay of number theory and arithmetic geometry, may be studied using methods from numerical algebraic geometry, when the base field is a transcendental extension of the complex numbers. This is because the well-known observation (which goes back to Hermite) that in this case Galois=monodromy, and computing monodromy is a basic operation in numerical algebraic geometry. While simply computing monodromy enables the exploration of a Galois group, it can only determine the group when it is the full symmetric group, for there is no stopping criterion. In work with Hauenstein and Rodriguez we offer two methods to determine a Galois group. For the first, we compute the branch locus, which leads to permutations that generate the Galois group. The second uses numerical irreducible decomposition of fiber products to determine the action of the Galois group, from which it may be determined.