The emerging applications of collaborative robots (cobots) are spilling out from product manufactories to service industries for human care, such as patient care for combating the coronavirus disease 2019 (COVID-19) pandemic and in-home care for coping with the aging society. There are urgent demands on equipping cobots with safe collaboration, immersive teleoperation, affective interaction, and other features (e.g., energy autonomy and self-learning) to make cobots capable of these application scenarios. Robot skin, as a potential enabler, is able to boost the development of cobots to address these distinguishing features from the perspective of multimodal sensing and self-contained actuation. This review introduces the potential applications of cobots for human care together with those demanded features. In addition, the explicit roles of robot skin in satisfying the escalating demands of those features on inherent safety, sensory feedback, natural interaction, and energy autonomy are analyzed. Furthermore, a comprehensive review of the recent progress in functionalized robot skin in components level, including proximity, pressure, temperature, sensory feedback, and stiffness tuning, is presented. Results show that the codesign of these sensing and actuation functionalities may enable robot skin to provide improved safety, intuitive feedback, and natural interfaces for future cobots in human care applications. Finally, open challenges and future directions in the real implementation of robot skin and its system synthesis are presented and discussed.