This work offers a historical account of the beginnings of medicine in Classic Greece, starting from Hippocrates, whom the Hellenes themselves, both in Antiquity and till our own days, acknowledge as the father of medical art. The article dwells on the exposition of the troublesome historiographical fact that the Corpus Hippocraticum is the key source available regarding the knowledge of western medicine's first steps, which has traditionally been called "the Hippocratic question". With a complete bibliographical annotation, this article invites also the qualitative evaluation of the successes achieved by ancient medicine in the fields of medical science as well as in those of anthropology and ethics.