Humans appear to be able to learn new concepts without needing to be programmed explicitly in any conventional sense. In this paper we regard learning as the phenomenon of knowledge acquisition in the absence of explicit programming. We give a precise methodology for studying this phenomenon from a computational viewpoint. It consists of choosing an appropriate information gathering mechanism, learning protocol, and exploring the class of concepts that can be learned using it in a reasonable (polynomial) number of steps. Although inherent algorithmic complexity appears to set serious limits to the range of concepts that can be learned, we show that there are some important nontrivial classes of propositional concepts that can be learned in a realistic sense.