This study is part of a larger research project on "Primary Care and the Construction of the Unified National Health System: Citizenship, Rights, and Public Policies", which takes a socio-historical perspective to question the Family Health Strategy. The article discusses the inclusion/exclusion of Primary Care for community demands and those related to issues not traditionally covered by health, such as persons with disabilities or mental disorders, with a focus on action by community health workers (CHWs). The study highlights the role of student training and the possibility of strengthening the integration between teaching and services for the demands by these population groups. Three experiences were analyzed in the cities of São Paulo and São Carlos, seeking to define strategies for awareness-raising and training of CHWs to deal with community demands. Real involvement with the community and awareness-raising and training of CHWs concerning the needs of specific population groups and social demands beyond basic health needs are fundamental strategies for implementing the notion of territorial responsibility in health and genuinely universal care.