Advances in whole-genome and whole-transcriptome amplification have permitted the sequencing of the minute amounts of DNA and RNA present in a single cell, offering a window into the extent and nature of genomic and transcriptomic heterogeneity which occurs in both normal development and disease. Single-cell approaches stand poised to revolutionise our capacity to understand the scale of genomic, epigenomic, and transcriptomic diversity that occurs during the lifetime of an individual organism. Here, we review the major technological and biological breakthroughs achieved, describe the remaining challenges to overcome, and provide a glimpse into the promise of recent and future developments.