An apparatus for processing data includes a processor operable in a plurality modes including at least one secure mode being a mode in a secure domain and at least one non-secure mode being a mode in a non-secure domain. When the processor is executing a program in a secure mode the program has access to secure data which is not accessible when the processor is operating in a non-secure mode. The processor is responsive to one or more exception conditions for triggering exception processing using an exception handler. The processor is operable to select the exception handler from among a plurality of possible exception handlers in dependence upon whether the processor is operating in the secure domain or the non-secure domain.