Null holomorphic curves in $\mathbb C^3$ and the conformal Calabi-Yau problem

In this paper we survey some recent contributions by the authors to the theory of null holomorphic curves in the complex Euclidean space $\mathbb C^3$, as well as their applications to null holomorphic curves in the special linear group $SL_2(\mathbb C)$, minimal surfaces in the Euclidean space $\mathbb R^3$, and constant mean curvature one surfaces (Bryant surfaces) in the hyperbolic 3-space $\mathbb H^3$. The paper is an expanded version of the lecture given by the second named author at the Abel Symposium in Trondheim, Norway, in July 2013.