Dissecting the circle, at random∗

Random laminations of the disk are the continuous limits of random non-crossing configurations of regular polygons. We provide an expository account on this subject. Initiated by the work of Aldous on the Brownian triangulation, this field now possesses many characters such as the random recursive triangulation, the stable laminations and the Markovian hyperbolic triangulation of the disk. We will review the properties and constructions of these objects as well as the close relationships they enjoy with the theory of continuous random trees. Some open questions are scattered along the text.