Thin, flexible robots able to bend and elongate can help surgeons reach deeper and more accurately into the human body than ever before, through increasingly smaller incisions. This talk will cover recent breakthroughs in design, control, and sensing that are rapidly pushing the boundaries of surgical robotics to smaller scales, greater accuracy, and more effective interaction with surgeons. Mechanics-based models of elastic robots provide the basis for these advancements, which in turn provide the raw materials necessary for building effective surgical robotic systems. These systems can offer autonomous, teleoperated, or hand-held surgeon-robot interactions. An important theme of the talk will be the fascinating process of partnering with surgeons to create new robots amenable to use in real-world operating room environments that have the potential to be powerful weapons in the fight against lung disease, brain tumors, hemorrhagic stroke, epilepsy, deafness, and urologic disorders.