  Over the past decade, the class of Oka manifolds has emerged from Gromov's
seminal work on the Oka principle. Roughly speaking, Oka manifolds are complex
manifolds that are the target of "many" holomorphic maps from affine spaces.
They are "dual" to Stein manifolds and "opposite" to Kobayashi-hyperbolic
manifolds. The prototypical examples are complex homogeneous spaces, but there
are many other examples: there are many ways to construct new Oka manifolds
from old. The class of Oka manifolds has good formal properties, partly
explained by a close connection with abstract homotopy theory. These notes were
prepared for lectures given at the Institute of Mathematics of the Chinese
Academy of Sciences in Beijing in May 2014. They are meant to give an
accessible introduction, not to all of Oka theory, but more specifically to Oka
manifolds, how they arise and what we know about them.
