In one-to-one Pickup and Delivery Problems (PDPs), the aim is to design a set of least cost vehicle routes starting and ending at a common depot in order to satisfy a set of pickup and delivery requests between location pairs, subject to side constraints. These requests apply to the transportation of goods or people, in which case the problem is often called the dial-a-ride problem. In recent years, there have been several significant developments in the area of exact and heuristic algorithms for PDPs. The purpose of this article is to report on these developments. It contains two main sections devoted to single vehicle and multi-vehicle problems, respectively. Each section is subdivided into two parts, one on exact algorithms and one on heuristics.