Human-robot collaboration in surgery represents a significant area of research, driven by the increasing capability of autonomous robotic systems to assist surgeons in complex procedures. This systematic review examines the advancements and persistent challenges in the development of autonomous surgical robotic assistants (ASARs), focusing specifically on scenarios where robots provide meaningful and active support to human surgeons. Adhering to the PRISMA guidelines, a comprehensive literature search was conducted across the IEEE Xplore, Scopus, and Web of Science databases, resulting in the selection of 32 studies for detailed analysis. Two primary collaborative setups were identified: teleoperation-based assistance and direct hands-on interaction. The findings reveal a growing research emphasis on ASARs, with predominant applications currently in endoscope guidance, alongside emerging progress in autonomous tool manipulation. Several key challenges hinder wider adoption, including the alignment of robotic actions with human surgeon preferences, the necessity for procedural awareness within autonomous systems, the establishment of seamless human-robot information exchange, and the complexities of skill acquisition in shared workspaces. This review synthesizes current trends, identifies critical limitations, and outlines future research directions essential to improve the reliability, safety, and effectiveness of human-robot collaboration in surgical environments.