In this paper we analyse the notion of collective responsibility and the criteria for its application to different types of groups. We argue that most of the ways in which the notion of collective responsibility has been attributed to different types of groups actually refer to a form of responsibility that is not genuinely collective, but that boils down to some form of individual responsibility. We identify an intrinsically collective kind of responsibility and argue that it can be attributed to only one kind of group. We begin by setting two necessary and sufficient conditions for attribution of genuinely collective moral responsibility, asking whether these two conditions are satisfied in the case of different types of groups that have been taken to be bearers of moral responsibility: organized groups, groups with internal bonds of solidarity, groups that program individuals to act in a certain way, random collections of individuals, and individuals engaging in joint actions. Contrary to what various authors have maintained, we argue that only in the case of individuals engaging in joint actions is attribution of a genuinely collective form of moral responsibility warranted, i.e. only groups engaging in joint action satisfy the two conditions for attribution of genuinely collective moral responsibility.