Important progress has been made in the study of the earthquake mechanism during the Upper Mantle Project, through the establishment of an appropriate mathematical framework which relates the observed seismogram with the slip motion across a fault plane. Since an arbitrary fault slip is described by a function of time and two space coordinates, a complete inversion of the observed seismogram is practically impossible. The only practical inversion method is to describe the kinematics of rupture growth along a fault plane using a small number of source parameters, and then to determine those parameters from the seismograms. A number of kinematic models have been proposed to cover a class of earthquakes. Preliminary attempts have also been made to find dynamic models by solving the problem of spontaneous rupture propagation for given initial conditions. There are a few source parameters, which represent some averaged quantities over the source time and source space and are therefore independent of the details of rupture kinematics. They are : (1) the seismic moment, which is proportional to the total displacement averaged over the fault surface; and(2) the apparent stress, which is proportional to the ratio of total seismic energy to seismic moment. In addition to these, the average stress drop can be approximately calculated from the seismic moment and the fault area, without detailed knowledge of the kinematics. The accuracies of determining these average quantities are discussed for the Parkfield, California earthquake of 1966, one of the best studied earthquakes during the U.M.P. period. Recent results from several well-studied earthquakes are then summarized as: (1) the relation between magnitude and seismic moment; and (2) the relation between stress-drop and magnitude. It was found that the stress drop in shallow earthquakes is 10-100 bar, apparently independent of magnitude for M > 6. Finally, recent evidences for the fault origin of deep-focus earthquakes are presented.