The resurgence of immune therapies in cancer medicine has elicited a corresponding interest in understanding the basis of patient response or resistance to these treatments. One aspect of patient response clearly lies in the genomic alterations that are associated with cancer onset and progression, including those that contribute to genomic instability and the resulting creation of novel peptide sequences that may present as neoantigens. The immune reaction to these unique ‘non-self’ peptides is frequently suppressed by the tumor itself, but the use of checkpoint blockade therapies, personalized vaccines, or a combination of these treatments may elicit a tumor-specific immune response that results in cell death. Massively parallel sequencing, coupled with different computational analyses, provides unbiased identification of the germline and somatic alterations that drive cancer development, and of those alterations that lead to neoantigens. These range from simple point mutations that change single amino acids to complex alterations, such as frameshift insertion or deletion mutations, splice-site alterations that lead to exon skipping, structural alterations that lead to the formation of fusion proteins, and other forms of collateral damage caused by genome instability that result in new protein sequences unique to the cancer. The various genome instability phenotypes can be identified as alterations that impact DNA replication or mismatch repair pathways or by their genomic signatures. This review provides an overview of current knowledge regarding the fundamentals of genome replication and of both germline and somatic alterations that disrupt normal replication, leading to various forms of genomic instability in cancers, to the resulting generation of neoantigens and, ultimately, to immune-responsive and resistant phenotypes.