Genocide and crimes against humanity are among the core crimes of international law, but they also carry great moral resonance due to their indissoluble link to the atrocities of the Nazi regime and to other egregious episodes of mass violence. However, the concepts of genocide and crimes against humanity are not well understood, even by the international lawyers and jurists who are most concerned with them. A conceptual fog hovers around the discussion of these two categories of crime. In this paper, I draw a number of distinctions aimed at clarifying the concepts. I distinguish three concepts of genocide, two legal and one moral, and two concepts of crimes against humanity, a legal and a moral one. I criticize the current legal concept of genocide and, using the idea of discrimination, propose a model for developing a more adequate legal concept and for better understanding the moral concept. I also criticize the moral concept of crimes against humanity, which many thinkers have conflated with the legal concept of such crimes.