The emergence of paired appendages is one of the most important adaptations of vertebrates; itprovided the opportunity to lead an active lifestyle and develop a wide range of ecological niches. Fish finsare a basic form of paired appendages in evolution. The problem of the emergence of paired appendages hasattracted the attention of researchers for more than 150 years. During this time, a number of theories havebeen proposed, based primarily on morphological data, two of which, the Balfour–Thatcher–Mivart lateralfold theory and the Gegenbaur gill arch theory, have not lost their relevance. However, to date, none of theproposed ideas has decisive evidence in its favor. In recent years, new data have been obtained in favor of eachof the theories, which emphasize the search for options for their consistent synthesis. In this review article,we attempt to consider the results and analyze the hypotheses regarding the prerequisites and mechanismsthat led to the emergence of paired appendages in vertebrates.