The CRISPR/Cas technology of targeted genome editing made it possible to carry out genetic engineering manipulations with eukaryotic genomes with a high efficiency. Targeted induction of site-specific DNA breaks is one of the key stages of the technology. The cell repairs the breaks via one of the two pathways, nonhomologous end joining (NHEJ) and homology-driven repair (HDR). The choice of the DNA repair pathway is determined by the architecture of the DNA break region formed as a result of terminal resection and depends on the cell cycle phase. NHEJ is the main pathway of double-strand break (DSB) repair in mammalian cells and involves a nonspecific ligation reaction. The reaction accuracy depends on the structure of break ends, and various insertions or deletions may arise as a result in the target genome region. Integration of a necessary sequence into the genome occurs via HDR, which requires a template with homology regions flanking a DSB. Introducing a genetic construct into a particular genomic locus is an important task, but is currently intricate and laborious to perform. However, the choice of the repair pathway can be of principal importance for basic research of gene functions and construction of animal models of human diseases to develop therapies. The review summarizes and systematizes the available information on strategies designed to increase the HDR efficiency. The strategies that most efficiently shift the balance towards HDR include use of NHEJ inhibitors, regulation of the key factors of homologous recombination, control of the cell cycle and chromatin status, and construction of HDR templates.