A general method of characteristics for solving the multigroup transport equation is developed. This is combined with an adaptive difference scheme, called the modified diamond scheme, and is then applied to the finite difference form of the equation. This formulation is obtained from the discrete ordinates equation, which in turn derives from the multigroup equation, both on the basis of consistency arguments. In this connection two forms of the multigroup equation are used, and the diffusion and other important limits also have a bearing on the final difference equation. The new approaches resolve a number of theoretical and practical difficulties with Sn-type transport calculations, in particular in curved and multidimensional geometries. They lead to a firmer basis for discrete ordinates quadrature sets and to better control, mesh cell by mesh cell, over flux extrapolation, including methods to smooth out unwanted flux oscillations. The total effect is a more consistent treatment of the transport equation together with improved accuracy, fewer breakdowns, and more speed in the calculations, while keeping close to the physics of the problem and retaining the basic simplicity of the difference approach.