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Problems of the theory of radio absorbers (RAs) involving frequency selective surfaces (FSSs) are considered. A design procedure for these RAs is described that takes into account the multiparameter character of the problem and allows one to determine the optimal characteristics of an FSS that provide the maximal operating bandwidth. RAs for a range of 1-17 GHz are obtained on the basis of polymer composites filled with carbonyl iron and Co2Z ferrite, into which an FSS in the form of a biperiodic array of thin metal rings is embedded. It is shown that the application of such an FSS allows one to increase the operating frequency bandwidth of a magnetic-type RA by a factor of more than 1.5 virtually almost without increasing the thickness of the absorber. Substantially, several types of FSS-based layered magnetodielectric RAs have been produced by compressing molding technique, including a 3-mm-thick RA with an operating bandwidth of 1.05-2.7 GHz, a 2-mm-thick RA with an operating bandwidth of 1.7-4.4 GHz, and a 1.6-mm-thick RA with an operating bandwidth of 6.7-16.1 GHz.