  Let $M$ be an open Riemann surface. We prove that every meromorphic function
on $M$ is the complex Gauss map of a conformal minimal immersion
$M\to\mathbb{R}^3$ which may furthermore be chosen as the real part of a
holomorphic null curve $M\to\mathbb{C}^3$. Analogous results are proved for
conformal minimal immersions $M\to\mathbb{R}^n$ for any $n>3$. We also show
that every conformal minimal immersion $M\to\mathbb{R}^n$ is isotopic through
conformal minimal immersions $M\to\mathbb{R}^n$ to a flat one, and we identify
the path connected components of the space of all conformal minimal immersions
$M\to\mathbb{R}^n$ for any $n\ge 3$.
