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Since the recent demonstration of chip-scale, silicon-based, photonic devices, silicon photonics provides a viable and promising platform for modern nonlinear optics. The development and improvement of such devices are helped considerably by theoretical predictions based on the solution of the underlying nonlinear propagation equations. In this paper, we review the approximate analytical tools that have been developed for analyzing active and passive silicon waveguides. These analytical tools provide the much needed physical insight that is often lost during numerical simulations. Our starting point is the coupled-amplitude equations that govern the nonlinear dynamics of two optical waves interacting inside a silicon-on-insulator waveguide. In their most general form, these equations take into account not only linear losses, dispersion, and the free-carrier and Raman effects, but also allow for the tapering of the waveguide. Employing approximations based on physical insights, we simplify the equations in a number of situations of practical interest and outline techniques that can be used to examine the influence of intricate nonlinear phenomena as light propagates through a silicon waveguide. In particular, propagation of single pulse through a waveguide of constant cross section is described with a perturbation approach. The process of Raman amplification is analyzed using both purely analytical and semianalytical methods. The former avoids the undepleted-pump approximation and provides approximate expressions that can be used to discuss intensity noise transfer from the pump to the signal in silicon Raman amplifiers. The latter utilizes a variational formalism that leads to a system of nonlinear equations that governs the evolution of signal parameters under the continuous-wave pumping. It can also be used to find an optimum tapering profile of a silicon Raman amplifier that provides the highest net gain for a given pump power.