The Newtonian two-body problem in R3, characterized by elliptical orbits, was generalized to S2. Vastly different phenomena may occur on S2 depending upon the initial separation of the two masses. Small separations, compared to the size of the universe, result in precessing ellipses whose rate of precession increases as the initial separation increases. When the initial separation is comparable to the size of the universe, complex orbits occur that are extremely sensitive to initial conditions, which is evidence of chaos. Similar behaviors are exhibited on S3. The equations of motion are integrated with a custom C++ program and with Mathematica.