TITLE: Why are Stein manifolds/spaces the analog of affine varieties/schemes in algebraic geometry?
QUESTION [15 upvotes]: I presume this is a GAGA-style result, but I cannot find a reference.

REPLY [24 votes]: Also like affine varieties, we have:

Theorem. A complex manifold is Stein if and only if it embeds into some $\mathbb{C}^N$ as a closed complex submanifold.

For the "only if" direction, see Hörmander, An Introduction to Complex Analysis in Several variables, Theorem 5.3.9. For the converse, an argument is contained on pp 109-110  of Hörmander, immediately after the definition of Stein manifold.