  Oka theory has its roots in the classical Oka principle in complex analysis.
It has emerged as a subfield of complex geometry in its own right since the
appearance of a seminal paper of M. Gromov in 1989. Following a brief review of
Stein manifolds, we discuss the recently introduced category of Oka manifolds
and Oka maps. We consider geometric sufficient conditions for being Oka, the
most important of which is ellipticity, introduced by Gromov. We explain how
Oka manifolds and maps naturally fit into an abstract homotopy-theoretic
framework. We describe recent applications and some key open problems. This
article is a much expanded version of the lecture given by the first-named
author at the conference RAFROT 2010 in Rincon, Puerto Rico, on March 22, 2010,
and of a brief survey article by the second-named author in Notices Amer. Math.
Soc., January 2010.
