I am aware that the theory of complex multiplication provides us with special functions whose values describe $K^\text{ab}$ where $K$ is an imaginary quadratic extension of the rationals. And that drinfeld modules provide a similarly explicit description of the maximal abelian extension of a global function field.
I am [vaguely] aware that the langlands programme would give us information about nonabelian extensions of global fields.
But what the problem of giving a concrete set of generators for abelian extensions of other fields $K$ when $K$ is not $\mathbb{Q}$, an imaginary quadratic number field, or a global function field? Why can't CM work on say, a real abelian number field? What goes wrong when you try to take the ideas of drinfeld modules back to an arbitrary number field?
Put another way - what is the state of Hilbert's 12th problem? Is it entirely subsumed by the langlands programme? And if not, what are the other proposed lines of attack?
As one out of many attempts at constructing class fields let me point out the work by Henri Darmon (see e.g. Elliptic Curves and Class Fields of Real Quadratic Fields: Algorithms and Evidence) on the construction of class fields of real quadratic number fields.