Stein manifolds, or affine analytic varieties, have been studied by complex
analysts for most of the past century. Their importance mandates the study
of two basic existence questions: Which smooth manifolds admit Stein
structures, and which open subsets of complex manifolds become Stein after isotopy
(using the complex structure inherited from the ambient space)? Building on
Eliashberg's fundamental work of the late 1980s, one can now completely
answer these questions. Eliashberg's surprisingly simple answer in high dimensions
extends to smooth 4-manifolds with an additional delicate condition.
Alternatively, one can eliminate the condition by working up to
homeomorphism and invoking Freedman theory. As an application, one obtains Stein open
subsets of C^2 realizing uncountably many diffeomorphism types of exotic R^4's.