The Oka-Grauert principle states that for any Stein manifold$X$ the holomorphic and the topological classification of complex vector bundles on $X$ coincide. The original reference is (Grauert 58).
The principle should maybe better be called the Oka-Grauert-Gromov principle/theory. Gromov viewed it in his book on partial differential relations as one of the examples of h-principle.
K. Oka, Sur les fonctions des plusieurs variables. III: Deuxième problème de Cousin, J. Sc. Hiroshima Univ. 9, 7–19 (1939)
H. Grauert, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135, 263–-273 (1958) doi
M. Gromov, Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989), 851–-897.
Finnur Lárusson, What is an Oka manifold, Notices AMS, pdf
Franc Forstnerič, Finnur Lárusson, Survey of Oka theory, arxiv/1009.1934
F. Forstnerič, The Oka principle for sections of stratified fiber bundles, Pure Appl. Math. Quarterly (Special Issue in honor of Joseph J. Kohn), 6 (2010), no. 3, 843–874, arxiv/0705.0591
There is now a model category structure on a category of presheaves of simplicial version of a Stein site where Oka maps are a fibration: