I have only just come across the remarkable theorem of Conway about universal quadratic forms over $\mathbb{Z}$; namely that in determining whether a integer coefficient, positive definite quadratic form represents all positive integers it suffices to check that it represents a specific set of 29 integers (the largest of which is 290). Other similar theorems exist too about representing all primes or all odd numbers.
I just read the paper by Bhargava and Hanke and found the proof quite elegant. I got thinking about whether it would generalize to other situations.
Has anyone has been able to extend the results to other settings? Maybe people have been able to prove similar things over other rings (such as rings of integers of number fields) or maybe people are still considering staying in the integer case and considering representing other sets of integers or considering higher degree forms?
Quite a bit of information is available as pdfs at my page TERNARY with what I hope are obvious names.
You want to look at Rouse on all odd numbers, ROUSE. Also Representation by ternary quadratic forms by OLIVER. In both cases some ineffective bounds are used, so a GRH is invoked that implies the suspected conclusions. This gives about the best conclusion to my paper with Kaplansky and Schiemann that I have any right to expect.
Hanke certainly thought that almost anything could be extended to integer rings of some number fields, and intended to find all class number one genera. This was an ambitious project, as it would require dimension up to 26. A student of Gabriele Nebe, named David Lorch, has found all positive class number one forms over $\mathbb Z,$ see LORCH.
I do not believe I know of any big-list papers on universality over number rings. There are some related approaches by Pete L. Cark of MO and MSE, see item 15 at CLARK. In this case, there was surely some influence by Hanke, who was at Georgia for some years.