The limitations on fisheries, the increasing demand for aquatic products, and the constant improvement of the biological knowledge on aquatic animals provide interesting opportunities for the domestication of new finfish species. The major historical landmarks of this recent activity are briefly described, emphasizing the cases of trout and salmon farming, yellowtail and marine species. Recent trends in temperate or tropical areas are described, discussing and addressing the major questions affecting finfish aquaculture development. Different aspects of the market (price, quantities), the evolution of product characteristics due to improved rearing practices and the consumer attitude are discussed through practical examples. Thus, diversification not only justifies significant research efforts, offering a solution for facing declining profits, but it has to be considered within a competitive context where some questions are finally addressed, i.e. the relation with the environment or the farming of introduced vs. endemic species.