We review recent advances in the DNA sequencing method based on measurements of transverse electrical currents. Device configurations proposed in the literature are classified according to whether the molecular fingerprints appear as the major (Mode I) or perturbing (Mode II) current signals. Scanning tunneling microscope and tunneling electrode gap configurations belong to the former category, while the nanochannels with or without an embedded nanopore belong to the latter. The molecular sensing mechanisms of Modes I and II roughly correspond to the electron tunneling and electrochemical gating, respectively. Special emphasis will be given on the computer simulation studies, which have been playing a critical role in the initiation and development of the field. We also highlight low-dimensional nanomaterials such as carbon nanotubes, graphene, and graphene nanoribbons that allow the novel Mode II approach. Finally, several issues in previous computational studies are discussed, which points to future research directions toward more reliable simulation of electrical current DNA sequencing devices.