As organisms age, cells accumulate genetic and epigenetic changes that eventually lead to impaired organ function or catastrophic transformation such as cancer. Since aging appears to be a stochastic process of increasing disorder1 cells in an organ will be individually affected in different ways - thus rendering bulk analyses of postmitotic adult tissues difficult to characterize. Here we directly measure the effects of aging in primary human tissue by performing single-cell transcriptome analysis of 2544 human pancreas cells from eight donors spanning six decades of life. We find that islet cells from older donors have increased levels of molecular disorder as measured both by noise in the transcriptome and by the number of cells which display inappropriate hormone expression, revealing a transcriptional instability associated with aging. By further analyzing the spectrum of somatic mutations in single cells, we found a specific age-dependent mutational signature characterized by C to A and C to G transversions. These mutations are indicators of oxidative stress and the signature is absent in single cells from human brain tissue or in a tumor cell line. We have used the single cell measurements of transcriptional noise and mutation level to identify molecular pathways correlated with these changes that could influence human disease. Our results demonstrate the feasibility of using single-cell RNA-seq data from primary cells to derive meaningful insights into the genetic processes that operate on aging human tissue and to determine molecular mechanisms coordinated with these processes.