Infants are highly curious and show remarkable self-driven learning capabilities. Inspired by developmental psychology and recent advances in neuroscience, computational models of curiosity have been implemented in robots. These models are based on the paradigm that learning progress generates intrinsic rewards, which in turn determine action selection. With the rise of deep learning, robots’ perceptual and behavioral learning capabilities have facilitated the appearance of infant-like curiosity-driven behaviors. Implemented in simulation, humanoid robots, drones and aquatic autonomous robots, these curiosity-based models enable open-ended hierarchical learning of skills, which can then be used for extrinsically formulated tasks. In this short review, we highlight the basic components of the most recent curiosity-based models, as well as their implementations in robots that learn about their own body, efficiently map their environment, explore object manipulation and tool use and socially engage with other agents. We conclude with remarks on future directions and challenges.