New breeding technologies have not only revolutionized biological science, but have also been employed to generate transgene-free products. Genome editing is a powerful technology that has been used to modify genomes of several important crops. This review describes the basic mechanisms, advantages and disadvantages of genome editing systems, such as ZFNs, TALENs, and CRISPR/Cas. Secondly, we summarize in detail all studies of the CRISPR/Cas system applied to potato and other tuber crops, such as sweet potato, cassava, yam, and carrot. Genes associated with self-incompatibility, abiotic-biotic resistance, nutrient–antinutrient content, and post-harvest factors targeted utilizing the CRISPR/Cas system are analyzed in this review. We hope that this review provides fundamental information that will be useful for future breeding of tuber crops to develop novel cultivars.