A competing risks framework refers to multiple risks acting simultaneously on a subject. A cure rate postulates a fraction of the subjects to be cured or failure-free, and can be formulated as a mixture model, or alternatively by a bounded cumulative hazard model. We develop models that unify the competing risks and cure rate approaches. The identifiability of these unified models is studied in detail. We describe Bayesian analysis of these models, and discuss conceptual, methodological and computational issues related to model fitting and model selection. We describe detailed applications in survival data from breast cancer patients in the Surveillance, Epidemiology, and End Results (SEER) program of the National Cancer Institute (NCI) of the United States.