The open-ended trap is one of the installations used for the magnetic confinement of thermonuclear plasma. Open-ended traps have a number of important advantages as compared with other confinement systems: they are attractive from the engineering point of view, their magnetic field is efficiently used to confine the plasma, they can be operated under steady-state conditions, and there is no particular problem with removing thermonuclear reaction products and heavy impurities from the plasma. At the same time, it has long been considered that the open-ended trap has a doubtful future as a basis for a thermonuclear reactor because of the relatively high rate of loss of plasma along the magnetic lines of force. The situation has changed for the better during the last decade, and a number of improved traps, that are largely free from this defect, has been proposed. This review examines the physical principles of open-ended traps (ambipolar, centrifugal, multimirror, gas-dynamic, and so on), the present state of research into these systems, and their future prospects. The use of open-ended traps as high-flux generators of 14-MeV neutrons is also discussed.