  Oka theory has its roots in the classical Oka-Grauert principle whose main
result is Grauert's classification of principal holomorphic fiber bundles over
Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds
and Stein spaces to Oka manifolds. 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.
  In this expository paper we discuss Oka manifolds and Oka maps. We describe
equivalent characterizations of Oka manifolds, the functorial properties of
this class, and geometric sufficient conditions for being Oka, the most
important of which is Gromov's ellipticity. We survey the current status of the
theory in terms of known examples of Oka manifolds, mention open problems and
outline the proofs of the main results.
  In the appendix by F. Larusson it is explained how Oka manifolds and Oka
maps, along with Stein manifolds, fit into an abstract homotopy-theoretic
framework.
  The article is an expanded version of the lectures given by the author at the
Winter School KAWA-4 in Toulouse, France, in January 2013. A more comprehensive
exposition of Oka theory is available in the monograph F. Forstneric, Stein
Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex
Analysis), Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 56,
Springer-Verlag, Berlin-Heidelberg (2011).
