Robots that carry out tasks and interact in complex environments will inevitably commit errors. Error detection is thus an important ability for robots to master, to work in an efficient and productive way. People leverage social cues from others around them to recognize and repair their own mistakes. With advances in computing and AI, it is increasingly possible for robots to achieve a similar error detection capability. In this work, we review current literature around the topic of how social cues can be used to recognize task failures for human-robot interaction (HRI). This literature review unites insights from behavioral science, human-robot interaction, and machine learning, to focus on three areas: 1) social cues for error detection (from behavioral science), 2) recognizing task failures in robots (from HRI), and 3) approaches for autonomous detection of HRI task failures based on social cues (from machine learning). We propose a taxonomy of error detection based on self-awareness and social feedback. Finally, we leave recommendations for HRI researchers and practitioners interested in developing robots that detect (physical) task errors using social cues from bystanders.