  In this paper we survey recent developments in the classical theory of
minimal surfaces in Euclidean spaces which have been obtained as applications
of both classical and modern complex analytic methods; in particular, Oka
theory, period dominating holomorphic sprays, gluing methods for holomorphic
maps, and the Riemann-Hilbert boundary value problem. Emphasis is on results
pertaining to the global theory of minimal surfaces, in particular, the
Calabi-Yau problem, constructions of properly immersed and embedded minimal
surfaces in $\mathbb{R}^n$ and in minimally convex domains of $\mathbb{R}^n$,
results on the complex Gauss map, isotopies of conformal minimal immersions,
and the analysis of the homotopy type of the space of all conformal minimal
immersions from a given open Riemann surface.
