In 1995 (if I'm not mistaken) Taylor and Wiles proved that all semistable elliptic curves over $\mathbb{Q}$ are modular. This result was extended to all elliptic curves in 2001 by Breuil, Conrad, Diamond, and Taylor.

I'm asking this as a matter of interest. Are there any other fields over which elliptic curves are known to be modular? Are there any known fields for which this is not true for? 

Also, is much research being conducted on this matter?