This book describes collisions between atoms that have been cooled to extremely low temperatures by optical and evaporative cooling techniques. The author reviews the elements of the quantum theory of scattering, and summarizes the theory and experimental techniques of optical cooling and trapping. Later chapters describe applications to precision spectroscopy, the determination of atomic properties, control of inelastic collisions by laser fields, and the manipulation of Bose-Einstein condensates (mesoscopic quantum systems). Finally, it reviews the essential properties of these mesoscopic quantum systems and describes the key importance of the scattering length to condensate stability. Descriptions of experiments and theory are included.