The so-called “neurological criterion” for the medical ascertainment of death (also called “brain death” or, better, “encephalic death”) has been widely accepted by the international scientific community and the majority of moralists and bioethicists. There is no shortage, however, of authors who take a contrary view. Their arguments and criticisms stand on different levels, which must be clearly distinguished. The subject has been addressed on several occasions by the Pontifical Academy of Sciences and other bodies linked in various ways to the universal Catholic Magisterium. The course intends to offer an accurate view of the different positions on this delicate issue, analysing the arguments of one side and the other, the available scientific data and the necessary anthropological and ethical reflections.