Two of the most classical theorems in the theory of holomorphic functions are the Runge approximation theorem and the Weierstrass interpolation theorem. In higher dimensions these correspond to the Oka-Weil approximation theorem and the Cartan extension theorem. A complex manifold X is said to be an Oka manifold if these classical results, and some of their natural extensions, hold for holomorphic maps from any Stein manifold (in particular, from complex Euclidean spaces) to X. After a brief historical review, beginning with the classical Oka-Grauert theory and continuing with the seminal work of Gromov, I will describe some recent developments and future challenges in this field of complex geometry. In particular, I shall describe a recently discovered connection between Oka theory and the classical theory of minimal surfaces in Euclidean spaces.