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This monograph deals with Burgers' equation and its generalisations. Such equations describe a wide variety of nonlinear diffusive phenomena, for instance, in nonlinear acoustics, laser physics, plasmas and atmospheric physics. The Burgers equation also has mathematical interest as a canonical nonlinear parabolic differential equation that can be exactly linearised. It is closely related to equations that display soliton behaviour and its study has helped elucidate other such nonlinear behaviour. The approach adopted here is applied mathematical. The author discusses fully the mathematical properties of standard nonlinear diffusion equations, and contrasts them with those of Burgers' equation. Of particular mathematical interest is the treatment of self-similar solutions as intermediate asymptotics for a large class of initial value problems whose solutions evolve into self-similar forms. This is achieved both analytically and numerically.