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Using finite difference methods this paper shows how solutions may be obtained with the aid of a digital machine to a wide range of microwave circuit problems. These problems include the parameters of TEM-mode transrnission lines, the equivalent circuits of obstacles in these lines, the cutoff frequencies of the fundamental mode in a waveguide of very general cross section, and the equivalent circuits of obstacles in rectangular waveguide. Methods for deriving the appropriate finite difference equations are presented and optimum methods for their solution set out; singularities are also included in the treatment. The paper ends with a resume of some typical results to problems of practical interest which have been obtained by these methods.