Cancer-associated mutations have been documented in normal tissues, but the prevalence and nature of somatic copy number alterations and their role in tumor initiation and evolution is not well understood. Here, using single cell DNA sequencing, we describe the landscape of CNAs in >42,000 breast epithelial cells from women with normal or high risk of developing breast cancer. Accumulation of individual cells with one or two of a specific subset of CNAs (e.g. 1q gain and 16q, 22q, 7q, and 10q loss) is detectable in almost all breast tissues and, in those from BRCA1 or BRCA2 mutations carriers, occurs prior to loss of heterozygosity (LOH) of the wildtype alleles. These CNAs, which are among the most common associated with ductal carcinoma in situ (DCIS) and malignant breast tumors, are enriched almost exclusively in luminal cells not basal myoepithelial cells. Allele-specific analysis of the enriched CNAs reveals that each allele was independently altered, demonstrating convergent evolution of these CNAs in an individual breast. Tissues from BRCA1 or BRCA2 mutation carriers contain a small percentage of cells with extreme aneuploidy, featuring loss of TP53 , LOH of BRCA1 or BRCA2 , and multiple breast cancer-associated CNAs in addition to one or more of the common CNAs in 1q, 10q or 16q. Notably, cells with intermediate levels of CNAs are not detected, arguing against a stepwise gradual accumulation of CNAs. Overall, our findings demonstrate that chromosomal alterations in normal breast epithelium partially mirror those of established cancer genomes and are chromosome- and cell lineage-specific.