Nanopore techniques offer a low-cost, label-free, and high-throughput platform that could be used in single-molecule biosensing and in particular DNA sequencing. Since 2010, graphene and other two-dimensional (2D) materials have attracted considerable attention as membranes for producing nanopore devices, owing to their subnanometer thickness that can in theory provide the highest possible spatial resolution of detection. Moreover, 2D materials can be electrically conductive, which potentially enables alternative measurement schemes relying on the transverse current across the membrane material itself and thereby extends the technical capability of traditional ionic current-based nanopore devices. In this review, we discuss key advances in experimental and computational research into DNA sensing with nanopores built from 2D materials, focusing on both the ionic current and transverse current measurement schemes. Challenges associated with the development of 2D material nanopores toward DNA sequencing are further analyzed, concentrating on lowering the noise levels, slowing down DNA translocation, and inhibiting DNA fluctuations inside the pores. Finally, we overview future directions of research that may expedite the emergence of proof-of-concept DNA sequencing with 2D material nanopores.