This review examines the dichotomy between automatic and autonomous behaviors in surgical robots, maps the possible levels of autonomy of these robots, and describes the primary enabling technologies that are driving research in this field. It is organized in five main sections that cover increasing levels of autonomy. At level 0, where the bulk of commercial platforms are, the robot has no decision autonomy. At level 1, the robot can provide cognitive and physical assistance to the surgeon, while at level 2, it can autonomously perform a surgical task. Level 3 comes with conditional autonomy, enabling the robot to plan a task and update planning during execution. Finally, robots at level 4 can plan and execute a sequence of surgical tasks autonomously. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 4 is May 3, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.