The inverse Galois problem studies which finite groups appear as the Galois group of some extension of the rational numbers. This talk will focus on a refinement of that problem: which finite groups appear as the Galois group of some extension of the rational numbers in which only a prescribed set of primes may ramify. I will provide results and examples in some specific cases in which only one finite prime ramifies, as well as remark on how the ramified primes in a Galois extension relate to generating sets of the corresponding Galois group.