There's alredy two posts on MO about the extension of modularity to elliptic curves over fields other than $\mathbb{Q}$ ([1], [2]), and another one about general algebraic varieties [3].

What is known about modularity of abelian varieties over $\mathbb{Q}$? I guess the arbitrary field case is at least as problematic as it is for elliptic curves.

Can be Serre's conjecture (now a theorem) generalized to more general representations $\rho:Gal(\bar{\mathbb{Q}}/\mathbb{Q}) \to GL_n(\bar{\mathbb{F_p}})$? Is there a Ribet-type result that that would imply full modularity?