In the past few years, our understanding of earth structure and of earthquakes has increased dramatically because of our greatly increased ability to interpret seismograms. Most of this progress can be attributed to the development of the necessary techniques for numerically synthesizing seismograms on a computer. Each of the processes that affect the waveform such as the seismic source or propagation of the wave through the earth is modeled by a linear time dependent operator. These linear operators are then convolved together to reproduce an analog waveform which can be compared directly to the data. In this arrangement, it is very simple to determine the nature and the magnitude of the effect that the various operators have on the waveform. The direct P and S body phases have proved to be the most useful types of seismic wave for modeling to date. A series of source studies based on body wave data has recently been completed. It has led to a completely new outlook on the relative importance of the rupture process versus the interaction of the seismic wave field with the earth's free surface in determining the body waveform. We shall review these studies in some detail. These source studies produced reliable source models which were then used to study the effects of the velocity structure of the upper mantle and of the shallow velocity structure near the source. We shall also discuss these results. Analogous progress which will not be covered in this review has also been made on modeling surface waves with time domain synthetics. Because of the much greater wavelengths involved, this line of research has not led to the type of detailed information about earthquakes and the earth which has come from body waves. It has led, nonetheless, to important evidence concerning the amount of slow deformation that accompanies some earthquakes and it appears to be much more useful for studying very large earthquakes. Important work in this field has been presented by Kanamori & Stewart (1976) and Gilbert & Dziewonski (1975).