Genomic analysis provides insights into the role of copy number variation in disease, but most methods are not designed to resolve mixed populations of cells. In tumours, where genetic heterogeneity is common, very important information may be lost that would be useful for reconstructing evolutionary history. Here we show that with flow-sorted nuclei, whole genome amplification and next generation sequencing we can accurately quantify genomic copy number within an individual nucleus. We apply single-nucleus sequencing to investigate tumour population structure and evolution in two human breast cancer cases. Analysis of 100 single cells from a polygenomic tumour revealed three distinct clonal subpopulations that probably represent sequential clonal expansions. Additional analysis of 100 single cells from a monogenomic primary tumour and its liver metastasis indicated that a single clonal expansion formed the primary tumour and seeded the metastasis. In both primary tumours, we also identified an unexpectedly abundant subpopulation of genetically diverse 'pseudodiploid' cells that do not travel to the metastatic site. In contrast to gradual models of tumour progression, our data indicate that tumours grow by punctuated clonal expansions with few persistent intermediates.