Monoclonal antibodies (mAbs) are the most promising and most intensively replenished type of bioactive pharmaceuticals. Currently, there are over 100 different mAbs approved by the FDA and other regulating agencies for treatment of oncological, infectious, systemic, autoimmune and other diseases. Design of antibodies neutralizing pathogens of socially significant infections, such as HIV, hepatitis viruses, SARS-CoV-2, is a separate direction. The SARS-CoV-2 pandemic has shown how urgent it is to have a technological platform enabling production of fully human antibodies. The development of recombinant DNA technology and antibody phage display enabled compilation of libraries of antigen-binding fragments and screening with target antigens. This review discusses the advantages and disadvantages of phage display, including use of single-domain antibody technology based on the heavy chain variable domain. We describe the state-of-the-art (and practical results of its application) technology enabling production of human antibodies by sorting and sequencing the genome of individual memory B cells, using monoclonal virus-neutralizing antibodies against SARS-CoV-2 as an example. The prospects of further development of the recombinant human antibody production technology are discussed; in particular, we consider creation of sequences of variable fragments of antibodies with the help of artificial intelligence.