The development of new catalysts with improved properties for organic reactions is relevant both for solving fundamental chemical challenges and for their implementation into production. In industry, acetylene hydrogenation is a catalytic process. The role of a catalyst is to increase the selectivity of hydrogenation to ethylene and prevent overhydrogenation to ethane. Research in the field of selective hydrogenation of acetylene, in addition to the applied relevance, is also of academic interest, since it discovers the nature of catalysis in details. Amazingly, acetylene hydrogenation has become, in a sense, a criterion for assessing the properties of synthesized catalysts, demonstrating the "structure–properties" relationship. Changes in the structure of a catalyst immediately result in changes in its catalytic characteristics, which, in turn, led to a more complete understanding of the processes occurring on the catalyst surface. A sharp increase in the number of researches on this subject that occurred in the period from 2021 to 2024 was due to the possibilities of obtaining and describing single-atom catalysts with the identification of the key dependence "particle size–activity". This review not only complements previous summarizing works on selective hydrogenation of acetylene, but also differs significantly in the arrangement of the material. Various approaches to the design of catalysts for this reaction are discussed based on the insight into its pathway. Mono- and bimetallic catalysts based on active metals such as palladium, nickel, copper, etc. are considered. Particular attention is paid to the application of single-atom catalysts. At the conclusion of this review, we summarized the catalytic characteristics of various systems in a table. The bibliography includes 267 references.