We will first review several heuristics on the distributions of Galois groups of unramified extensions of global fields, which include the Cohen-Lenstra Heuristics regarding the class groups of quadratic fields and the Boston-Bush-Hajir Heuristics regarding the p-class tower groups of quadratic fields. We will then discuss how these heuristics relate to reasonable random group models, and then explain a new conjecture on the distribution of the Galois groups of the maximal unramified extensions of Galois Γ number fields or function fields for a large family of finite groups Γ. Finally, we will give theorems in the function field case to support this new conjecture. This work is joint with Melanie Matchett Wood and David Zureick-Brown. Speaker(s): Yuan Liu (University of Michigan)