The merged-beams technique is powerful for the experimental study of certain classes of atomic and molecular processes that cannot be as readily or accurately addressed by other methods. The principal advantages of the technique are the ability to make quantitative studies of collisional interactions with high resolution at low relative energies, to collect products that have undergone appreciable angular scattering, and to investigate processes involving short-lived or chemically-reactive species. Despite continuing advances in ion-source and particle-beam technologies, merged-beams experiments remain a challenge, constituting a relatively small but growing fraction of the worldwide effort in atomic and molecular collisions research. This review outlines the fundamental principles of the merged-beams method, reviews techniques and progress, and focuses on three active programs to highlight the advantages of the method for addressing fundamental questions in atomic and molecular physics.