The solution of Hilbert's tenth problem by Davis, Matiyasevich, Putnam, and Robinson, showing that the recursively enumerable sets are precisely the Diophantine sets.

(I am reproducing here a blog post, Hindsight, from a couple of years ago, as suggested in the comments.)

From Georg Kreisel's review of The decision problem for exponential diophantine equations, by Martin Davis, Hilary Putnam, and Julia Robinson, Ann. of Math. (2), 74 (3), (1961), 425–436. MR0133227 (24 #A3061).

This paper establishes that every recursively enumerable (r.e.) set can be existentially defined in terms of exponentiation. […] These results are superficially related to Hilbert’s tenth problem on (ordinary, i.e., non-exponential) Diophantine equations. The proof of the authors’ results, though very elegant, does not use recondite facts in the theory of numbers nor in the theory of r.e. sets, and so it is likely that the present result is not closely connected with Hilbert’s tenth problem. Also it is not altogether plausible that all (ordinary) Diophantine problems are uniformly reducible to those in a fixed number of variables of fixed degree, which would be the case if all r.e. sets were Diophantine.

Of course, my favorite quote in relation to the tenth problem is from the Foreword by Martin Davis to Yuri Matiyasevich’s Hilbert’s tenth problem.

During the 1960s I often had occasion to lecture on Hilbert’s Tenth Problem. At that time it was known that the unsolvability would follow from the existence of a single Diophantine equation that satisfied a condition that had been formulated by Julia Robinson. However, it seemed extraordinarily difficult to produce such an equation, and indeed, the prevailing opinion was that one was unlikely to exist. In my lectures, I would emphasize the important consequences that would follow from either a proof or a disproof of the existence of such an equation. Inevitably during the question period I would be asked for my own opinion as to how matters would turn out, and I had my reply ready: “I think that Julia Robinson’s hypothesis is true, and it will be proved by a clever young Russian.”