Citizen science encompasses activities with scientific objectives in which members of the public participate as more than passive research subjects from whom personal data or biospecimens are collected and analyzed by others. Citizen science is increasingly common in the biomedical sciences, including the fields of genetics and human genomics. Genomic citizen science initiatives are diverse and involve citizen scientists in collecting genetic data, solving genetic puzzles, and conducting experiments in community laboratories. At the same time that genomic citizen science is presenting new opportunities for individuals to participate in scientific discovery, it is also challenging norms regarding the manner in which scientific research outputs are managed. In this review, we present a typology of genomic citizen science initiatives, describe ethical and legal foundations for recognizing genomic citizen scientists’ claims of credit for and control of research outputs, and detail how such claims are or might be addressed in practice across a variety of initiatives.