Yes, but not like a solid would move. The bits and boulders of Saturn’s rings each orbit like tiny moons. They move around Saturn in the same direction as Saturn rotates and the same direction as all the known moons (except the outermost moon, Phoebe, which orbits in the opposite direction). The ring particles closest to Saturn whiz around the fastest, and those farther away travel around at a slower speed (in keeping with Kepler’s Laws). The orbital speed of a ring particle depends only on its distance from Saturn’s center; it does not depend on the particle’s mass or size. For example, if a house-sized ring particle and a sugar grain–sized ring particle are both orbiting at the inner edge of the B ring, they move at the same speed around Saturn. However, ring particles at the outer edge of the B ring orbit at a slower speed. The particles of the innermost rings are moving around Saturn faster than Saturn rotates. Ring particles can also move randomly in other directions, such as perpendicular to the rings. These motions are usually damped out quickly by collisions or gravitational interaction between the particles. If you wait a much longer time (millions of years), some of the rings may spread out and eventually disappear.