Evolutionary biology has not suceeded in defining a concept of species that will satisfy all researchers. This critical survey of the main proposals and their respective difficulties tends, on the one hand, to throw light on the processes of concept formation in the empirical sciences, and, on the other, to undermine the paratheological vision of knowledge and truth that initially inspired modern science and still prevails among many educated persons. The article is divided into two parts. The first part concerns the biological (or genetical) concept of species which was adopted by Theodosius Dobzhansky and Ernst Mayr ca. 1940 and some alternatives which were subsequently introduced to overcome its limitations. The second part deals with several branches of the cladist tradition founded by Willi Hennig (1950). Various concepts of species that could not be easily integrated in either group were omitted for the sake of coherence and brevity.