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This paper proposes a new approach to magnetic recording based on shingled writing and two-dimensional readback and signal-processing. This approach continues the use of conventional granular media but proposes techniques such that a substantial fraction of one bit of information is stored on each grain. Theoretically, areal-densities of the order of 10 Terabits per square inch may be achievable. In this paper we examine the feasibility of this two-dimensional magnetic recording (TDMR) and identify the significant challenges that must be overcome to achieve this vision.