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In this review article, we describe group-III nitride laser diodes that emit light in the green spectral range, using epitaxial structures grown on gallium nitride (GaN) substrates with c- and semipolar-plane orientations. We address the motivation for these lasers, the challenges faced in creating them, and the progress made in this field to date. Different structural design choices are described, taking into account specific material properties and crystal growth requirements for these orientations. We review various properties of the materials involved, including optical gain, optical confinement, internal optical losses and carrier injection. We also discuss mechanical strain during the growth of active and passive regions, and the way in which it limits the structural design. Various aspects of laser chip fabrication are discussed, including self-aligned ridge waveguides and facet formation. Finally, we outline the status of green laser reliability and challenges in this area.