The behavior of linear hyperbolic waves has been analyzed by decomposing the waves into pieces in space-time and into different frequencies. The linear nature of the equations involved allows the reassembling of the pieces in a simple fashion; the individual pieces do not interact. For nonlinear waves the interaction of the pieces seemed to preclude such an analysis, but in the late 1970s it was shown that a similar procedure could be undertaken in this case and would yield important information. The analysis of the decomposed waves, and of waves with special smoothness or size in certain directions, has been fruitful in describing a variety of the properties of nonlinear waves. This volume presents a number of articles on topics of current interest which involves the use of the newer techniques on nonlinear waves. The results established include descriptions of the smoothness of such waves as determined by their geometry, the properties of solutions with high frequency oscillations, and the long-time smoothness and size estimates satisfied by nonlinear waves.