Gauss famously investigated class numbers of quadratic fields, in particular characterizing the 2-divisibility of the class number for such fields. In general, it is expected that for a number field of any degree, and any rational prime p, the p-torsion part of the class group  should be arbitrarily small, in a suitable sense, relative to the absolute discriminant of the field. This talk will present recent progress for both quadratic and higher degree number fields.