TITLE: Instances where an existence result precedes the constructive version
QUESTION [22 upvotes]: 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!

REPLY [6 votes]: Hironaka's original desingularization algorithm had constructive aspects but as I understand, also had aspects / computations which are not constructive and could not be done by hand or with a computer.  
Resolution algorithms now typically are based upon blowing up the strata of some invariant (where heuristically the singularities are at their worst) and so they are constructive if you can compute the invariant.  But these invariants might be for all intents and purposes completely incomputable, for example based on infinite amounts of data.
New algorithms have simplified many aspects on this, and indeed, algorithms have been implemented in computers, and it isn't so hard to write down all the various invariants as you follows modern algorithms. 
You can see 
Hironaka desingularisation theorem -- new proofs in literature?
for more discussion.