We investigate a class of feature allocation models that generalize the Indian buffet process and are parameterized by Gibbs-type random measures. Two existing classes are contained as special cases: the original two-parameter Indian buffet process, corresponding to the Dirichlet process, and the stable (or three-parameter) Indian buffet process, corresponding to the Pitman--Yor process. Asymptotic behavior of the Gibbs-type partitions, such as power-laws holding for the number of latent clusters, translates into analogous characteristics for this class of Gibbs-type feature allocation models. We derive a number of properties of these models, present a black-box inference procedure, and elucidate the subtle differences between this superclass and its better known subclasses.