In this paper we introduce a new class of Pickup and Delivery problems on circles (or rings). These problems arise in the field of public transportation systems where autonomous (i.e. driverless) vehicles travel on circular networks. We consider a set of stations arranged in a circle and a set of transportation requests. Each request asks for the transportation of a certain quantity from a pickup station to a delivery station. A fleet of capacitated vehicles is available at the depot. In the first part of the paper we propose a classification scheme for these problems. In the second part, we address the variants in which the vehicles are allowed to move in a single direction of the circle (either clockwise or counterclockwise) and the objective is to minimize the number of tours on the ring while serving all the requests. We provide a complexity analysis for this class of problems. We develop polynomial time algorithms for the variants that are polynomially solvable and proofs of NP-hardness for the variants that are NP-hard. In addition, for the latter, we provide mathematical formulations and perform computational tests that show the effectiveness of these formulations. Finally, we compare optimal solutions with those obtained using a straightforward greedy algorithm.