The basic motivation here is to encourage and inspire - via examples - the pursuit of alternate proofs of existing results that might be more accessible and intuitive by cataloging success stories. Here's the question:
Are there good examples of instances in mathematics research where an existence-only result preceded - by some number of years - the corresponding constructive result?
I am aware of one nice example from my own field, which I hope is illustrative of the type of answer that would be nice to collect here: Tucker's Lemma, which is a discrete version of the famous Borsuk-Ulam Theorem, was first proved in the following paper by contradiction:
A. W. Tucker. Some topological properties of disk and sphere. In Proc. First Canadian Math. Congress, Montreal, 1945, pages 285–309. University of Toronto Press, Toronto, 1946.
The first constructive proof (which is starkly different from the original), did not appear in the literature until
R. M. Freund and M. J. Todd. A constructive proof of Tucker’s combinatorial lemma. J. Combin. Theory Ser. A, 30(3):321–325, 1981.
I did try just searching google for "A constructive proof of" and similar search strings. This does provide some examples, but the results are not filtered by importance of the result in question, or the extent of difference between the old existence result and the newer constructive one.
Only one example per answer, please!