A family of spatial discretization formulas, based on piecewise Chebyshev expansions with C0 continuity, is given for the solution of a general class of parabolic equations. These formulas are obtained by first expressing the generalized Chebyshev method of Berzins & Dew (1981) in a Galerkin framework, and then using this framework to simplify the method. An analysis of the new and old discretization formulas is given and a comparison made with a finite-difference method. A method is described for obtaining an indication of the error in the numerical solution that takes account of both the spatial and temporal approximations.