The possibility of group minds or group mental states has been considered by a number of authors addressing issues in social epistemology and related areas (Goldman 2004, Pettit 2003, Gilbert 2004, Hutchins 1995). An appeal to group minds might, in the end, do indispensable explanatory work in the social or cognitive sciences. I am skeptical, though, and this essay lays out some of the reasons for my skepticism. The concerns raised herein constitute challenges to the advocates of group minds (or group mental states), challenges that might be overcome as theoretical and empirical work proceeds. Nevertheless, these hurdles are, I think, genuine and substantive, so much so that my tentative conclusion will not be optimistic. If a group mind is supposed to be a single mental system having two or more minds as proper parts, the prospects for group minds seem dim–or so I will argue.