Abstract: We give a complete description of nonlinear waves and their pairwise interactions in isentropic gas dynamics. Our analysis includes rarefactions, compressions and shock waves. Because the waves are arbitrarily large, we describe the change of states across the wave exactly, without resolving the characteristic patterns. We similarly describe the states between nonlinear waves in any pairwise interaction. When the strengths and reflected waves are described correctly, we show that whenever two (arbitrary) nonlinear waves of the same family interact, their strengths simply add. Also, if a wave crosses a simple wave (rarefaction or compression) of the opposite family, its strength is unchanged, and the change in the opposite simple wave is explicitly given by exact formulae. In addition, we analyze the crossing of two arbitrary shocks. We obtain bounds for the outgoing middle state, and use this to estimate the outgoing wave strengths in terms of the incident strengths. Our estimates are global in that they apply to waves of arbitrary strength, and they are uniform in the incoming middle state. In particular, the estimates continue to hold as the middle state approaches vacuum.