It is shown that the species concept of Ernst Mayr does not consider the evolution and modes of reproduction of eucaryotic organisms as a whole. It is only translatable into a taxonomic practice in a very special situation: sexually reproducing and sympatrically occurring organisms that do not exchange genes. Mayr’s central criterion of reproductive isolation is not applicable to the pervasive cases of reticulate evolution, to numerous groups of organisms with asexual reproduction, to the frequent situations of allopatry and to classification of fossil organisms. Evaluating advantages and disadvantages of five broadly applied species concepts and integrating elements of the related concepts of Sonneborn (1957), Sokal & Crovello (1970) and De Queiroz (2007), a new synthesis, called the Pragmatic Species Concept, is presented: ‘A species is a cluster of organisms which passed a threshold of evolutionary divergence. Divergence is determined by one or several operational criteria described by an adequate numerics. A single conclusive operational criterion is sufficient. Conflicts between operational criteria require an evolutionary explanation. Thresholds for each operational criterion are fixed by consensus among the experts of a discipline under the principle of avoiding over-splitting. Clusters must not be the expression of intraspecific polymorphism.’ This concept is applicable to all known groups of eukaryotic organisms independent from their mode of reproduction or evolutionary history. It allows both an approach by multisource integrative taxonomy as well as by a single discipline and is open for integrating new disciplines. The concept enables sound taxonomic decisions also in case of reticulate evolution, parthenogenesis, apomixis, allopatry, separate time horizons and reversal of strong evolutionary divergence. The complex problem could only be solved by focusing on the degree of evolutionary divergence, reproducible numeric data recording, adequate numeric analyses and the threshold principle. Recommendations of how to translate this concept into a taxonomic practice are given.