It's strange to see an article about the progress of modularity conjectures for elliptic curves that doesn't mention either Taniyama or Shimura, and instead mentions Langlands as the originator of the idea. This is bizarre historical revisionism.
Taniyama and Shimura only formulated a conjecture about elliptic curves over the rational numbers. They didn't come up with a statement about elliptic curves over number fields, and this article is about work to generalize modularity to number fields. The Langlands program, on the other hand does, help enormously with such generalizations. It's not trivial to formulate a correct conjectural generalization of modularity of elliptic curves over general number fields, and some naive analogues don't work at all...