We consider the problem of reaching agreement in distributed systems in which some processes may deviate from their prescribed behavior before they eventually crash. We call this failure model "mortal Byzantine". After discussing some application examples where this model is justified, we provide matching upper and lower bounds on the number of faulty processes, and on the required number of rounds in synchronous systems. We then continue our study by varying different system parameters. On the one hand, we consider the failure model under weaker timing assumptions, namely for partially synchronous systems and asynchronous systems with unreliable failure detectors. On the other hand, we vary the failure model in that we limit the occurrences of faulty steps that actually lead to a crash in synchronous systems.