The problem of analyzing the number of number field extensions $L/K$ with bounded (relative) discriminant has been the subject of renewed interest in recent years, with significant advances made by Ellenberg-Venkatesh, Kable-Yukie, and (especially) Bhargava. I will give an overview of the history of this problem and what results are known (or conjectured), and then discuss my work on a series of generalizations, using similar techniques to Ellenberg-Venkatesh, for giving an upper bound on the number of extensions $L/K$ with fixed degree, bounded relative discriminant, and specified Galois closure.