Drugs and treatments could be precisely tailored to an individual patient by extracting their cellular-and molecular-level information. For this approach to be feasible on a global scale, however, information on complete genomes (DNA),transcriptomes (RNA) and proteomes (all proteins) needs to be obtained quickly and at low cost. Quantum mechanical phenomena could potentially be of value here, because the biological information needs to be decoded at an atomic level and quantum tunnelling has recently been shown to be able to differentiate single nucleobases and amino acids in short sequences. Here, we review the different approaches to using quantum tunnelling for sequencing, highlighting the theoretical background to the method and the experimental capabilities demonstrated to date. We also explore the potential advantages of the approach and the technical challenges that must be addressed to deliver practical quantum sequencing devices.