TITLE: Applications of Stein spaces in Algebraic Geometry
QUESTION [7 upvotes]: I want to know where are essential applications of the theory of Stein spaces in algebraic geometry. I heard Cartan's theorem A & B were used in Serre's GAGA, but are there any other applications?

REPLY [7 votes]: I would say that technically speaking Stein spaces are essentially not used in algebraic geometry but that they had, and still have, a strong motivational role. Let me elaborate on these polemical assertions.    
First of all algebraic geometry, and especially scheme theory, does not presuppose that the base field is $\mathbb C$ and even less that it has characteristic zero. So Stein manifolds can't even be defined. 
Stein manifolds are replaced in algebraic geometry by affine varieties (or schemes), and theorems for them are infinitely easier than the corresponding theorems for Stein spaces: the vanishing of cohomology for coherent sheaves (and even for quasi-coherent sheaves!) is proved in two pages over general affine noetherian schemes (not only varieties) in Hartshorne, whereas the corresponding theorem for Stein spaces was developed over two decades by Oka, Cartan, Serre, ... and takes half a book to be proved (cf. the book by Grauert-Remmert on Stein spaces or that by L.Kaup-B.Kaup on holomorphic functions of several variables).    
And finally the empirical evidence is overwhelming: of the numerous algebraic geometry books which handle cohomology of coherent sheaves over an algebraic variety exactly zero (to my knowledge) use Stein manifolds to prove their results.    
That said, historically the first use of the cohomology of coherent sheaves was indeed in complex manifold theory : Cartan realized that Oka's results on pseudo-convex domains could be interpreted in the language of Leray's recently discovered  sheaves and Cartan's student Serre adapted these techniques to algebraic geometry in his ground-breaking article Faisceaux Algébriques Cohérents.
But let me emphasize again that Serre never uses any result from Stein theory in that article.