  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.
