As the title says. In particular, every elliptic curve over $\mathbb{Q}$ is modular; but what is the current state of the art for general totally real number fields? I assume the answer is extractable from some papers of, say, Kisin, but I am not an expert in this material and hesitate to try that myself.

Elliptic curves over real quadratic fields were proven to be modular very recently by Freitas, Le Hung and Siksek:
Some progress was alredy present in the thesis of Richard Taylor's student Le Hung.
Potentially modular is known unconditionally over arbitrary totally real fields, that is:
As mentioned in the comments, for practical purposes this is almost as good as modularity. The attribution of this theorem is complicated. There's some information about this in section 7 of our own Kevin Buzzard's very interesting survey on modularity:
You can also consult:
For arbitrary totally real fields, modularity is still not known. 

