I'm surprised no one has mentioned Freedman's Theorem from 1982 yet. Technically this theoerem says that Casson handles are standard, but more broadly this completes the classification of topological 4-manifolds, and is one of two major theorems going in to the proof of the existence of exotic 4-manifolds.

(The other half is Donaldson's work on gauge theories. That half has been somewhat simplified with the introduction of Seiberg-Witten theory.)

I have never attempted to read his proof, but it at least has a reputation for being hard.