This paper defends a 'noetic' conception of aesthetic experience whereby such experience is best conceived as a kind of explorative thought process. Although not directly aimed at acquiring knowledge, this process often leads to an enhanced understanding or improved epistemic grasp of the object of appreciation itself and the world. On this conception, aesthetic value acts as an invitation to engage in a series of contemplative and reflective processes during which we rely not only on the perceptual, imaginative, and affective abilities which have occupied such a central role in aesthetic theory, but also on our capacities for sense-making, problem-solving and theory-building. Cases of intelligible beauty or aesthetic value should thus lie at the heart of accounts of aesthetic experience.