Due to an increase in the consumption of food, feed, and fuel and to meet global food security needs for the rapidly growing human population, there is a necessity to obtain high-yielding crops that can adapt to future climate changes. Currently, the main feed source used for ruminant livestock production is forage grasses. In temperate climate zones, perennial grasses grown for feed are widely distributed and tend to suffer under unfavorable environmental conditions. Genome editing has been shown to be an effective tool for the development of abiotic stress-resistant plants. The highly versatile CRISPR-Cas system enables increasingly complex modifications in genomes while maintaining precision and low off-target frequency mutations. In this review, we provide an overview of forage grass species that have been subjected to genome editing. We offer a perspective view on the generation of plants resilient to abiotic stresses. Due to the broad factors contributing to these stresses the review focuses on drought, salt, heat, and cold stresses. The application of new genomic techniques (e.g., CRISPR-Cas) allows addressing several challenges caused by climate change and abiotic stresses for developing forage grass cultivars with improved adaptation to the future climatic conditions. Genome editing will contribute towards developing safe and sustainable food systems.