TITLE: Examples of "miraculous" proofs
QUESTION [39 upvotes]: Concerning the proof that $\zeta(3)$ is irrational, Van der Poorten famously noted that
"Apéry's incredible proof appears to be a mixture of miracles and mysteries".
Indeed, many ideas introduced in Apéry's proof such as $\zeta(3)=\frac{5}{2} \sum_{k=1}^{\infty}{\frac{ (-1)^{k-1}} {\binom {2k}{k}k^{3}}}$ and the recurrence $n^3u_n + (n-1)^3 u_{n-2} = (34n^3-51n^2+27n-5)u_{n-1}, n\geq2$ amazed contemporary mathematicians (although the fast-converging series was already derived several years earlier by Hjortnaes).
What are other examples of "miraculous" proofs, whose ingredients amazed mathematicians of their time? In particular, proofs such that:

a large portion of the Theorems involved in the proof represent entirely new ideas,
these Theorems are applicable to a wide area of mathematics,
the general atmosphere among contemporary mathematicians was a mixture of  surprise and awe ("where did this come from?").

REPLY [4 votes]: I am surprised that nobody has mentioned the Riemann hypothesis and its consequences. I can never hide my fascination that somebody could manage to prove that the distribution of prime numbers and many other problems in number theory, all boil down to proving that the roots of a seemingly nice function in the complex plane have equal real parts.
I wish I could some day travel back in time and ask him really, what kind of sorcery is this?