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Two-dimensional magnetic recording (TDMR) differs from traditional track recording in that the bit-size-to-grain-size ratio is drastically reduced so that the channel bits are roughly of the same sizes as those of the magnetic media grains. This is envisioned to be achieved by shingled writing, a write-process in which the corner-writer partially overwrites the previously written adjacent track, effectively writing very closely spaced narrow tracks with no guard bands. Since the tracks are very narrow, they can be read by an array of read-elements whose spacing is equal to the narrow track pitch, creating a two-dimensional (2-D) array of readback signals (the two dimensions being the cross-track and the down-track dimensions). In the absence of an array of read elements, the same 2-D readback can be obtained by progressive scans of a single read element that reads one narrow track at a time and stores the readback signal in a 2-D array. In this paper, we present simple magnetic grain media models and shingled-write process models that capture the essence of TDMR. We assume that the granular recording medium consists of randomly shaped tiles (each tile represents a grain), randomly covering the medium plane. We then derive a suitable 2-D read/write process model. Using proper information-theoretic inequalities and bounding techniques, we derive methods to bound the capacities of TDMR channels. The results reveal that information capacities in excess of 0.6 user bits per grain are possible to attain over TDMR channels.