building on work of jordan from 1870, in 1979 harris showed that a geometric monodromy group associated to a problem in enumerative geometry is equal to the galois group of an associated field extension. vakil gave a geometric-combinatorial criterion that implies a galois group contains the alternating group. these galois groups are difficult to determine, yet they contain subtle geometric information. exploiting harris's equivalence, leykin and i used numerical homotopy continuation to compute galois groups of problems involving mostly divisor schubert classes, finding all to be the full symmetric group. (this included one problem with 17589 solutions.) with brooks and martin del campo, we used vakil's criterion to show that all schubert problems involving lines have at least alternating galois group. my talk will describe this background and sketch a current project to systematically determine galois groups of all schubert problems of moderate size on all small classical flag manifolds, investigating at least several million problems. this will use supercomputers employing several overlapping methods, including combinatorial criteria, symbolic computation, and numerical homotopy continuation, and require the development of new algorithms and software.