Chromosomal instability (CIN) and epigenetic alterations are characteristics of advanced and metastatic cancers^1-4, but whether they are mechanistically linked is unknown. Here we show that missegregation of mitotic chromosomes, their sequestration in micronuclei^5,6 and subsequent rupture of the micronuclear envelope⁷ profoundly disrupt normal histone post-translational modifications (PTMs), a phenomenon conserved across humans and mice, as well as in cancer and non-transformed cells. Some of the changes in histone PTMs occur because of the rupture of the micronuclear envelope, whereas others are inherited from mitotic abnormalities before the micronucleus is formed. Using orthogonal approaches, we demonstrate that micronuclei exhibit extensive differences in chromatin accessibility, with a strong positional bias between promoters and distal or intergenic regions, in line with observed redistributions of histone PTMs. Inducing CIN causes widespread epigenetic dysregulation, and chromosomes that transit in micronuclei experience heritable abnormalities in their accessibility long after they have been reincorporated into the primary nucleus. Thus, as well as altering genomic copy number, CIN promotes epigenetic reprogramming and heterogeneity in cancer.