Gene therapy is the straightforward approach for the application of recent advances in molecular biology into clinical practice. One of the major obstacles in the development of gene therapy is the delivery of the effector to and into the target cell. Unfortunately, most methods commonly used in laboratory practice are poorly suited for clinical use. Viral vectors are one of the most promising methods for gene therapy delivery. Millions of years of evolution of viruses have resulted in the development of various molecular mechanisms for entry into cells, long-term survival within cells, and activation, inhibition, or modification of the host defense mechanisms at all levels. The relatively simple organization of viruses, small genome size, and evolutionary plasticity allow modifying them to create effective instruments for gene therapy approaches. This review summarizes the latest trends in the development of gene therapy, in particular, various aspects and prospects of the development of clinical products based on viral delivery systems.