This review, which is the first of two, focuses on the male reproductive anatomy and reproductive technologies used in Neo-tropical hystricomorphic rodents with the potential for domestication, which are the agouti (Dasyprocta leporina), the capybara (Hydrochoerus hydrochaeris) and the paca (Cuniculus paca). We consider over seventy references spanning from 1965 to 2020, with the majority of work being done in the past twenty years. Knowledge of the reproductive tract and reproductive technologies is critical to the conservation and preservation of these species. Although all three animals had similarities in their anatomy, such as no overt scrotums and testes located intra-abdominally in the inguinal region, some had unique features—for example, the agouti and the paca had penile spines, and two lateral penile cartilages. High spermatogenic efficiency was noted in the agouti and the paca, making them good candidates for increasing their reproductive performance in conservation programs. A review of the literature has shown that there is increasing work taking place on the reproductive technologies used in these animals; however, a lot of work is still lacking, as, to the author’s knowledge, standard protocols and artificial insemination procedures are yet to be established.