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The genitalia of many male insects include structures whose functions are unknown or poorly understood. The endophallus of many Lepidoptera bears sclerotized structures known as cornuti, which in some species break off during copulation and remain within the female genital tract (“deciduous” cornuti). I describe previous and original hypotheses on the role of cornuti, identify the selective pressures invoked by these hypotheses, propose different ways of testing them and briefly review pertinent evidence. I describe ten functional hypotheses for non-deciduous cornuti and four for deciduous cornuti; six hypotheses invoke natural selection and eight involve sexual selection. In some cases more than one of the proposed functions could be performed by cornuti; evolutionary change from one function to another is also possible. I suggest that the wide morphological variation observed in non-deciduous …