In a recent work with A.Alarcón and I.Castro-Infantes we show that every open Riemann surface admits a complete conformal CMC-1 (constant mean curvature one) immersion in the three dimensional hyperbolic space. In this talk I aim to explain the main ideas in the proof of this result, which relies on the holomorphic representation of CMC-1 surfaces given by Robert Bryant in 1987, and applies modern complex analysis techniques.