Neutrons provide an important tool for investigations in physics. Ultracold neutrons (UCN) belong to the extremely low energy range (10]-7 eV to 10-8 eV), have velocities in the range 1-10 m s-1 and wavelengths of 100-1000 A. Ultracold neutrons can be kept in hermetically sealed vessels forup to 15 minutes until they decay. This time is sufficient to observe the action of very weak fields and gives an insight into the properties of neutrons and their environment. Capture and heating by wall nuclei reduce the storage time of neutrons to below 15 minutes. Present research areas include attempts to understand and prevent the large losses of UCN, improve storage time, and to obtain fundamental results such as an upper limit to the electric dipole moment ofneutrons and improved measurements of decay time. These studies have also led to a new approach to diffraction and diffusion theory, which is applicable to any radiation: waves or corpuscles. The book explains how physicists should cope with the problems of UCN research, how UCN can be used forfundamental and applied research, and summarizes the results which have been published.