Work in Human-Robot Interaction (HRI) has investigated interactions between one human and one robot as well as human-robot group interactions. Yet, the field lacks a clear definition and understanding of the influence a robot can exert on interactions between other group members (e.g., human-to-human). In this paper, we define Interaction-Shaping Robotics (ISR), a subfield of HRI that investigates robots that influence the behaviors and attitudes exchanged between two (or more) other agents. We highlight key factors of Interaction-Shaping Robots that include the role of the robot, the robot-shaping outcome, the form of robot influence, the type of robot communication, and the timeline of the robot’s influence. We also describe three distinct structures of human-robot groups to highlight the potential of ISR in different group compositions and discuss targets for a robot’s interaction-shaping behavior. Finally, we propose areas of opportunity and challenges for future research in ISR.