Understanding the absolute Galois group of number fields is one of the most important questions in number theory. While this problem is extremely difficult in full generality, one can gain insight by studying quotients of absolute Galois group corresponding to extensions satisfying additional conditions (for example, unramified, tamely ramified, or pro-$p$ extensions). In this talk, we will discuss results on presentations of Galois groups of global fields with restricted ramification, and their applications to the study of the distribution of class groups of given families of global fields.