We calculate the energy states and Aharonov-Bohm oscillations of an electron in elliptical quantum rings in the presence of a uniform magnetic field by using an exact numerical diagonalization. The calculated results show that the elliptical quantum rings are flatter, larger amplitudes and periods of the Aharonov-Bohm oscillations are observed. In addition, in the limits of a circular quantum ring, the results of our approach are in good agreement with those of earlier theories.