Hodge theory (after Deligne) In an interview with Deligne on the Simons Foundation website, I heard Robert MacPherson say that at the time Deligne's papers on Hodge theory were being published, the results seemed absolutely miraculous to specialists in Algebraic Geometry. 
Can anyone explain why this is ? 
 A: A brief answer. 
First of all, the results are miraculous. Deligne's Hodge II and Hodge III give just a few example applications of the kind of results you can prove using mixed Hodge theory; these are great theorems which just fall out of the general theory. People quickly figured out more applications, like the Hodge-Deligne polynomial, which is itself a complete miracle - and its existence just falls out of the general theory.
But there is also the fact that the results seemingly came out of nowhere. In fact Deligne was motivated by the philosophy of motives, which was not very well known/understood at the time outside the initiated few around Grothendieck. To my knowledge the only accounts of motives in the literature at the time were Kleiman's paper in the Oslo proceedings and Manin's paper on the blow-up formula for Chow motives, and it would be a long time until anyone talked about "mixed motives". The weight filtration appeared naturally in the $\ell$-adic cohomology when you considered a variety which was not necessarily smooth and projective, and the Frobenius eigenvalues on $\mathrm{Gr}^W_i H^k$ would be of absolute value $q^{i/2}$, so they would look like part of the degree $i$ cohomology of a smooth proper variety. And "by motivic philosophy" that phenomenon should correspond to something on the Hodge theory side. But without this motivic philosophy as guidance the weight filtration looks completely ad hoc and unmotivated.
A: The laudatio for the Wolf prize explains it like this:

Central to modern algebraic geometry is the theory of moduli, i.e.,
  variation of algebraic or analytic structure. This theory was
  traditionally mysterious and problematic. In critical special cases,
  i.e., curves, it made sense, i.e., the set of curves of genus greater
  than one had a natural algebraic structure. In dimensions greater than
  one, there was some sort of structure locally, but globally everything
  remained mysterious. [...] Building on Mumford’s and Griffiths’ work,
  Pierre Deligne demonstrated how to extend the variation of Hodge
  theory to singular varieties. This advance, called mixed Hodge theory,
  allowed explicit calculation on the singular compactification of
  moduli spaces that came up in Mumford’s geometric invariant theory.

