During the summer of 1976, the two Viking spacecraft, each consisting of an orbiter-lander combination, were inserted into orbit about Mars. The paper describes the experiences of the Viking Satellite Orbit Determination Team in determining Mars centered ephemerides of the orbiters and positions of the landers from the two-way Doppler and range data, and synthesizes the different phases of the navigation plan which involves pre-flight modeling and error analysis for all Viking navigation functions from launch through landing. The problem of initial orbit convergence is solved by using DPODP's (Double Precision Orbit Determination Program) square-root batch data filter, and gravity models for both Viking I and Viking II were produced from the combination of short arc estimates. The importance of synchronous Viking orbits (with the rotational period of Mars) is stressed and future extended missions of the spacecraft are outlined.