How do humans come to know so much about the world from so little data? Even young children can infer the meanings of words, the hidden properties of objects, or the existence of causal relations from just one or a few relevant observations -- far outstripping the capabilities of conventional learning machines. How do they do it -- and how can we bring machines closer to these human-like learning abilities? I will argue that people's everyday inductive leaps can be understood in terms of (approximations to) probabilistic inference over generative models of the world. These models can have rich latent structure based on abstract knowledge representations, what cognitive psychologists have sometimes called "intuitive theories", "mental models", or "schemas". They also typically have a hierarchical structure supporting inference at multiple levels, or "learning to learn", where abstract knowledge may itself be learned from experience at the same time as it guides more specific generalizations from sparse data. This talk will focus on models of learning and "learning to learn" about categories, word meanings and causal relations. I will show in each of these settings how human learners can balance the need for strongly constraining inductive biases -- necessary for rapid generalization -- with the flexibility to adapt to the structure of new environments, learning new inductive biases for which our minds could not have been pre-programmed. I will also discuss briefly how this approach extends to richer forms of knowledge, such as intuitive psychology and social inferences, or physical reasoning. Finally, I will raise some challenges for our current state of understanding about learning in the brain, and neurally inspired computational models.