We present theoretical ab initio results for the elastic constants of two different phases of SiO2, stishovite and alpha-quartz using the plane-wave-pseudopotential method. Our calculated values are in very good agreement with the experimental data as well as some of the existing model calculations. SiO2 is one of the most abundant minerals in the earth's mantle. Therefore, the knowledge of its elastic behavior will be very important from a seismology point of view. Our calculated sound velocities are presented and are accordingly also in good agreement with those derived from experiment.