Laser microsurgery is the current gold standard surgical technique for the treatment of selected diseases in delicate organs such as the larynx. However, the operations require large surgical expertise and dexterity, and face significant limitations imposed by available technology, such as the requirement for direct line of sight to the surgical field, restricted access, and direct manual control of the surgical instruments. To change this status quo, the European project μRALP pioneered research towards a complete redesign of current laser microsurgery systems, focusing on the development of robotic micro-technologies to enable endoscopic operations. This has fostered awareness and interest in this field, which presents a unique set of needs, requirements and constraints, leading to research and technological developments beyond μRALP and its research consortium. This paper reviews the achievements and key contributions of such research, providing an overview of the current state of the art in robot-assisted endoscopic laser microsurgery. The primary target application considered is phonomicrosurgery, which is a representative use case involving highly challenging microsurgical techniques for the treatment of glottic diseases. The paper starts by presenting the motivations and rationale for endoscopic laser microsurgery, which leads to the introduction of robotics as an enabling technology for improved surgical field accessibility, visualization and management. Then, research goals, achievements, and current state of different technologies that can build-up to an effective robotic system for endoscopic laser microsurgery are presented. This includes research in micro-robotic laser steering, flexible robotic endoscopes, augmented imaging, assistive surgeon-robot interfaces, and cognitive surgical systems. Innovations in each of these areas are shown to provide sizable progress towards more precise, safer and higher quality endoscopic laser microsurgeries. Yet, major impact is really expected from the full integration of such individual contributions into a complete clinical surgical robotic system, as illustrated in the end of this paper with a description of preliminary cadaver trials conducted with the integrated μRALP system. Overall, the contribution of this paper lays in outlining the current state of the art and open challenges in the area of robot-assisted endoscopic laser microsurgery, which has important clinical applications even beyond laryngology.