In number theory, the Sato-Tate conjecture about elliptic curves over $ \mathbf Q$ was a problem from the 1960s and Serre's conjecture on modularity of odd 2-dimensional Galois representation was a conjecture from the 1970s-1980s. Both were settled around 2008. (For ST conj., the initial proof needed an additional technical hypothesis -- not part of the original conjecture -- of a non-integral $j$-invariant, which was later removed in 2011.) For those not familiar with these problems, their solutions build on ideas coming from the proof of Fermat's Last Theorem.