This paper gives asymptotic solutions to generalized Burgers’ equations governing the propagation of weakly nonlinear acoustic waves under the influence of geometrical spreading and thermoviscous diffusion. Geometrical effects are included through a general ray-tube area function, A(r), and solutions calculated to arbitrarily large ranges, for both N-wave and sinusoidal initial signals, in two limits, one in which a dimensionless diffusivity is allowed to vanish for fixed values of source Mach number and spreading parameter wr 0 /a 0? , the other in which the source Mach number is allowed to increase indefinitely for fixed values of the other parameters, the latter simulating experiments in which the phenomenon of amplitude saturation was investigated. At moderate ranges the wave pattern comprises the lossless portions, separated by shocks of Taylor structure, of weak-shock theory, and we pinpoint a number of non-uniformities in this description at large ranges, rectifying the non-uniformities with appropriate new asymptotic descriptions of the wave and classifying the area functions, A (r), accordingly. The validity of weak-shock theory at large ranges is delineated, and, where it fails, different routes to the final old-age linear decay are identified. Complete solutions in old age are obtained for a broad class of area functions including that for spherical waves. Amplitude saturation in the appropriate limit is shown to be a general phenomenon for these model equations. The paper ends with a discussion of various ad hoc approaches to approximate solution of generalized Burgers’ equations and with some comparison with experiment.