  This is an expanded version of my plenary lecture at the 8th European
Congress of Mathematics in Portoro\v{z} on 23 June 2021. The main part of the
paper is a survey of recent applications of complex-analytic techniques to the
theory of conformal minimal surfaces in Euclidean spaces. New results concern
approximation, interpolation, and general position properties of minimal
surfaces, existence of minimal surfaces with a given Gauss map, and the
Calabi-Yau problem for minimal surfaces. To be accessible to a wide audience,
the article includes a self-contained elementary introduction to the theory of
minimal surfaces in Euclidean spaces.
