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The design of media for ultrahigh-density perpendicular recording is discussed in depth. Analytical and semianalytical models are developed to determine the constraints upon the media to fulfill requirements of writability and thermal stability, and the effect of intergranular exchange coupling is examined. The role of vector fields during the write process is examined, and it is shown that one-dimensional models of perpendicular recording have significant deficiencies. A micromagnetic model is described and the results of simulations of recording undertaken with the model are presented. The paper demonstrates that there is no physical reason why perpendicular recording should not be possible at or above 1 Tb/in2.