What are the functions of curiosity? What are the mechanisms of curiosity-driven learning? We approach these questions about the living using concepts and tools from machine learning and developmental robotics. We argue that curiosity-driven learning enables organisms to make discoveries to solve complex problems with rare or deceptive rewards. By fostering exploration and discovery of a diversity of behavioural skills, and ignoring these rewards, curiosity can be efficient to bootstrap learning when there is no information, or deceptive information, about local improvement towards these problems. We also explain the key role of curiosity for efficient learning of world models. We review both normative and heuristic computational frameworks used to understand the mechanisms of curiosity in humans, conceptualizing the child as a sense-making organism. These frameworks enable us to discuss the bi-directional causal links between curiosity and learning, and to provide new hypotheses about the fundamental role of curiosity in self-organizing developmental structures through curriculum learning. We present various developmental robotics experiments that study these mechanisms in action, both supporting these hypotheses to understand better curiosity in humans and opening new research avenues in machine learning and artificial intelligence. Finally, we discuss challenges for the design of experimental paradigms for studying curiosity in psychology and cognitive neuroscience.