Meiosis, although essential for reproduction, is also variable and error-prone: rates of chromosome crossover vary among gametes, between the sexes, and among humans of the same sex, and chromosome missegregation leads to abnormal chromosome numbers (aneuploidy)1-8. To study diverse meiotic outcomes and how they covary across chromosomes, gametes and humans, we developed Sperm-seq, a way of simultaneously analysing the genomes of thousands of individual sperm. Here we analyse the genomes of 31,228 human gametes from 20 sperm donors, identifying 813,122 crossovers and 787 aneuploid chromosomes. Sperm donors had aneuploidy rates ranging from 0.01 to 0.05 aneuploidies per gamete; crossovers partially protected chromosomes from nondisjunction at the meiosis I cell division. Some chromosomes and donors underwent more-frequent nondisjunction during meiosis I, and others showed more meiosis II segregation failures. Sperm genomes also manifested many genomic anomalies that could not be explained by simple nondisjunction. Diverse recombination phenotypes-from crossover rates to crossover location and separation, a measure of crossover interference-covaried strongly across individuals and cells. Our results can be incorporated with earlier observations into a unified model in which a core mechanism, the variable physical compaction of meiotic chromosomes, generates interindividual and cell-to-cell variation in diverse meiotic phenotypes.