Why was sexual selection so important to Darwin? And why was it de-emphasized by almost all of Darwin's followers until the second half of the 20th century? These two questions shed light on the complexity of the scientific tradition named "Darwinism". Darwin's interest in sexual selection was almost as old as his discovery of the principle of natural selection. From the beginning, sexual selection was just another "natural means of selection", although different from standard "natural selection" in its mechanism. But it took Darwin 30 years to fully develop his theory, from the early notebooks to the 1871 book The Descent of Man, and Selection in Relation to Sex. Although there is a remarkable continuity in his basic ideas about sexual selection, he emphasized increasingly the idea that sexual selection could oppose the action of natural selection and be non adaptive. In time, he also gave more weight to mate choice (especially female choice), giving explicit arguments in favor of psychological notions such as "choice" and "aesthetic sense". But he also argued that there was no strict demarcation line between natural and sexual selection, a major difficulty of the theory from the beginning. Female choice was the main reason why Alfred Russel Wallace, the co-discoverer of the principle of natural selection, engaged in a major controversy with Darwin about sexual selection. Wallace was suspicious about sexual selection in general, trying to minimize it by all sorts of arguments. And he denied entirely the existence of female choice, because he thought that it was both unnecessary and an anthropomorphic notion. This had something to do with his spiritualist convictions, but also with his conception of natural selection as a sufficient principle for the evolutionary explanation of all biological phenomena (except for the origin of mind). This is why Wallace proposed to redefine Darwinism in a way that excluded Darwin's principle of sexual selection. The main result of the Darwin-Wallace controversy was that most Darwinian biologists avoided the subject of sexual selection until at least the 1950s, Ronald Fisher being a major exception. This controversy still deserves attention from modern evolutionary biologists, because the modern approach inherits from both Darwin and Wallace. The modern approach tends to present sexual selection as a special aspect of the theory of natural selection, although it also recognizes the big difficulties resulting from the inevitable interaction between these two natural processes of selection. And contra Wallace, it considers mate choice as a major process that deserves a proper evolutionary treatment. The paper's conclusion explains why sexual selection can be taken as a test case for a proper assessment of "Darwinism" as a scientific tradition. Darwin's and Wallace's attitudes towards sexual selection reveal two different interpretations of the principle of natural selection: Wallace's had an environmentalist conception of natural selection, whereas Darwin was primarily sensitive to the element of competition involved in the intimate mechanism of any natural process of selection. Sexual selection, which can lack adaptive significance, reveals this exemplarily.