The partially cooperative relay broadcast channel (RBC) is studied, where a source transmits to two users, and one user also acts as a rein y node to assist broadcast communication. The source has common information intended for both users and private information for each user. Inner and outer bounds on the capacity region are derived for discrete memoryless partially cooperative RBCs, These bounds are further refined for three example channels; partially cooperative RBCs with degraded message sets, semideterministic partially cooperative RBCs, and orthogonal partially cooperative RBCs. The inner and outer bounds are shown to match for the latter two channels, and hence the capacity regions are established. Furthermore, when the relay function is disabled, the inner and outer bounds reduce to potentially new bounds on the capacity region of broadcast channels.