Mutations, the fuel of evolution, are first manifested as rare DNA changes within a population of cells. Although next-generation sequencing (NGS) technologies have revolutionized the study of genomic variation between species and individual organisms, most have limited ability to accurately detect and quantify rare variants among the different genome copies in heterogeneous mixtures of cells or molecules. We describe the technical challenges in characterizing subclonal variants using conventional NGS protocols and the recent development of error correction strategies, both computational and experimental, including consensus sequencing of single DNA molecules. We also highlight major applications for low-frequency mutation detection in science and medicine, describe emerging methodologies and provide our vision for the future of DNA sequencing.