We present results of atomistic computer simulations of spontaneous and stress-induced grain boundary (GB) migration in copper. Several symmetrical tilt GBs have been studied using the embedded-atom method and molecular dynamics. The GBs are observed to spontaneously migrate in a random manner. This spontaneous GB motion is always accompanied by relative translations of the grains parallel to the GB plane. Furthermore, external shear stresses applied parallel to the GB and normal to the tilt axis induce GB migration. Strong coupling is observed between the normal GB velocity vn and the grain translation rate v||. The mechanism of GB motion is established to be local lattice rotation within the GB core that does not involve any GB diffusion or sliding. The coupling constant between vn and v|| predicted within a simple geometric model accurately matches the molecular dynamics observations.