Gastric cancer is a very serious and deadly disease worldwide with about one million new cases every year. Most gastric cancer subtypes are associated with genetic and epigenetic aberrations caused by chromosome instability, microsatellite instability or Epstein-Barr virus infection. Another risk factor is an infection with Helicobacter pylori, which also triggers severe alterations in the host genome. This pathogen expresses an extraordinary repertoire of virulence determinants that take over control of important host cell signaling functions. In fact, H. pylori is a paradigm of persistent infection, chronic inflammation and cellular destruction. In particular, H. pylori profoundly induces chromosomal DNA damage by introducing double-strand breaks (DSBs) followed by genomic instability. DSBs appear in response to oxidative stress and pro-inflammatory transcription during the S-phase of the epithelial cell cycle, which mainly depends on the presence of the bacterial cag pathogenicity island (cagPAI)-encoded type IV secretion system (T4SS). This scenario is closely connected with the T4SS-mediated injection of ADP-glycero-β-D-manno-heptose (ADP-heptose) and oncoprotein CagA. While ADP-heptose links transcription factor NF-κB-induced innate immune signaling with RNA-loop-mediated DNA replication stress and introduction of DSBs, intracellular CagA targets the tumor suppressor BRCA1. The latter scenario promotes BRCAness, a disease characterized by the deficiency of effective DSB repair. In addition, genetic studies of patients demonstrated the presence of gastric cancer-associated single nucleotide polymorphisms (SNPs) in immune-regulatory and other genes as well as specific pathogenic germline variants in several crucial genes involved in homologous recombination and DNA repair, all of which are connected to H. pylori infection. Here we review the molecular mechanisms leading to chromosomal DNA damage and specific genetic aberrations in the presence or absence of H. pylori infection, and discuss their importance in gastric carcinogenesis.