A coupled integral-equation method for calculating fields in complex cavities is developed and is applied to cavities comprising multiple cascaded and overlapping coaxial and circular-cylindrical regions or sections. The field in a section is related to the fields in apertures of planar conducting surfaces, which bound the section. Laboratory cavities are constructed, and their fields are computed from numerical solutions of the integral equations. The results are compared with data obtained through measurements in both the frequency and time domains. The accuracy of the method is also verified by comparison of measured and calculated values of the current and charge on cavity walls. The tools developed are used to investigate the salient features of a sine burst that passes through the complex cavity.