According to this report, Louis De Branges claims to have proved the Riemann Hypothesis. If correct, it’s very significant – much more so than the proof of Fermat’s Last Theorem by Wiles.
It is also, I think, the last of the big and well-known unsolved problems in mathematics, and its nice to see the search ending in success. Some of the other big problems have been closed, rather than solved. The classic problems of the Greeks such as squaring the circle were shown to be insoluble in the 19th century, and the Hilbert program of formalisation was shown by Godel to be infeasible. And the four-colour problem (not a really important problem, but a big one because it was easily described, interesting and very tough) was dealt with by a brute-force computer enumeration.
Almost instant update Commenter Eric points to Mathworld which says “Much ado about nothing”. On the other hand, the same page reports a proof of the infinitude of twin primes which has been an open question for a long time, though not a problem in the same league as those mentioned above.