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Let $M$ be an open Riemann surface. We prove that every meromorphic functionon $M$ 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 aholomorphic null curve $M\to\mathbb{C}^3$. Analogous results are proved forconformal minimal immersions $M\to\mathbb{R}^n$ for any $n>3$. We also showthat every conformal minimal immersion $M\to\mathbb{R}^n$ is isotopic throughconformal minimal immersions $M\to\mathbb{R}^n$ to a flat one, and we identifythe path connected components of the space of all conformal minimal immersions$M\to\mathbb{R}^n$ for any $n\ge 3$.

Every meromorphic function is the Gauss map of a conformal minimal surface