A nonlinear theory of internal gravity waves is extended to slowly varying mean winds u0. This theory determines the wave amplitudes, enhanced diffusion and momentum deposition, for a broad spectrum of waves as well as for a single wave. It is shown that the momentum deposition is not constant with altitude, even above the altitude where a wave saturates (breaks). The deposition is seen to decrease as a critical level is approached. The rate of decrease varies with spectral width. In general, the behavior of saturated waves near a critical level is shown to differ greatly from that of linearly growing waves. It is then proven that the momentum deposition is always related to the diffusion coefficient in a very simple and general way. Hence, knowledge of one implies knowledge of the other. Both quantities vary with altitude in nearly the same way. A feature of the theoretical momentum deposition is that it sometimes behaves like Rayleigh friction and sometimes not. It is referred to as “generalized Rayleigh friction,” and the different cases are delineated. It is pointed out that the terms “saturation altitude” or “wavebreak altitude” are ambiguous when applied to a broad spectrum because different wavelengths saturate at different altitudes. Hence, there is a range of saturation altitudes. If the spectrum is broad enough, saturation will occur over a wide range of altitudes, thereby causing diffusion and deposition throughout much of the middle atmosphere. By use or rocket data, the theoretical momentum deposition is estimated to be 40 m a day−1 near the stratopause (about 45 km) and more than 100 m s−1 day−1 in regions of the mesosphere. It is very small below 40 km altitude because the waves are linear there. The (vertical) wave diffusion coefficient is about 10 m2 s−1 just below the stratopause, more than 200 m2 s−1 in regions of the mesosphere, and very small below 40 km. In addition, the horizontal diffusion is shown to be much larger than the vertical, as was pointed out in a previous paper. It is also found that the diffusion and momentum deposition vary greatly with the spectral width, being much smaller for a very broad spectrum than for a narrow spectrum. The spectral width has a crucial affect on the height profiles of diffusion and deposition. The theory is compared to previous theories and to some observations.