As said in comments, the question has already been discussed on MO, see the links given there.

To summarize and complete, it is important to remember that in the case of number fields the correspondence between Galois representations and certain automorphic representations (the ones which are algebraic at infinity in the case of number fields) is only a part of the Langlands program, essentially because it concerns just certain automorphic representations, and Langlands functoriality is about all of them.

In the function field case, for $\operatorname{GL}_n$ over an arbitrary field, the correspondence has been completely done 15 years ago by Laurent Lafforgue. More recently (last year) Vincent Lafforgue (younger brother of Laurent) has done the sense Automorphic $\rightarrow$ Galois for an arbitrary reductive group (that is to an automorphic form he attaches a suitable $L$-parameter as conjectured, that is a kind of twisted Galas representation). In the introduction of his paper he seems optimistic about his prospect to solve (with Genestier) the converse
direction, which entails carefully regrouping the automorphic representations into classes called $L$-packets.

In the number field case, much less is known, but much more than 20 years ago. In the sense
Automorphic $\Rightarrow$ Galois, we can now do the case of automorphic representations for $\operatorname{GL}_n$ over a field with is either totally real or CM, and which are not only algebraic but *regular* at infinity. We can also do this for other groups (unitary, orthogonal, symplectic) but that gives no new Galois representations so I don't dwell on it (though in the proof, we need to do the case of these groups before going to $\operatorname{GL}_n$). The great final stone was put, after a huge collective effort leading to the case of self-dual or conjugate slef-dual representations, by Harris-Lan-Taylor-Thorne and then by another method by Scholze, which also deals with the case of torsion automorphic forms, not part of the initial Langlands program. The next main frontier seems to me to be able to deal with non-regular algebraic automorphic representations, the simplest case of which being algebraic Maass form for $\operatorname{GL}_2$.

Much progress has been made since Wiles and Taylor's proof of FLT on the converse direction Galois $\Rightarrow$ Automorphic, but it definitely lags behind the other sense. Essentially, due to work of Harris-Taylor and many others, the case of almost all conjugate self-dual Galois representation is done. Work in progress involving a numb or people could do many new cases, perhaps all where we know the automorphic to Galois sense.

This is essentially what is known about the Global Galois/Automorphic correspondence. For the functoriality for (non necessarily algebraic) automorphic representations, much less is known (Solvable base change for $\operatorname{GL}_n$ by Langlands and Arthur-Clozel, some transfer between classical groups, some low degree exterior powers) but much remains to be done.

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