https://ncatlab.org/nlab/show/Oka+principle#Forstneric12

says:

``In complex geometry, the Oka-Grauert principle states that over complex manifolds 
 which are Stein manifolds, the non-abelian cohomology-classification of holomorphic vector bundles coincides with that of topological vector bundles, .....This was originally proven for holomorphic line bundles in Oka 1913 (in which case it says that holomorphic line bundles over Stein manifolds are fully classified by their first Chern class) and generalized in Grauert 1958 to holomorphic vector bundles and further to holomorphic principal bundles with structure group any complex Lie group.
''

ALSO SEE: 

https://www.fmf.uni-lj.si/~forstneric/papers/2012Gromov%27scontribution.pdf

Gromov's redoing of the Oka-Grauert principle