Universal quadratic forms over number fields

I will talk about several recent results on universal quadratic forms over
rings of integers of totally real number fields (i.e., totally positive
quadratic forms that represent all totally positive integers). Over real
quadratic fields, one can obtain a fairly precise information concerning
the smallest rank of a universal form in terms of the associated continued
fraction; in particular, the rank can be arbitrarily large. Things are much
more complicated in the higher degree case, but I will also discuss some
partial results, e.g., related to the (non)existence of universal forms
whose coefficients are rational integers.