Hurwitz Monodromy and Full Number Fields

We give conditions for the monodromy group of a Hurwitz space over the configuration space of branch points to be the full alternating or symmetric group on the degree. Specializing the resulting coverings suggests the existence of many number fields with full Galois group and surprisingly little ramification --- for example, the existence of infinitely many such number fields unramified away from {2,3,5}.