An apparatus and method are provided for executing debug instructions. The apparatus has processing circuitry for executing instructions fetched from memory, and a debug interface. The processing circuitry is responsive to a halt event to enter a halted mode where the processing circuitry stops executing the instructions fetched from memory, and instead is arranged to execute debug instructions received from a debugger via the debug interface. The processing circuitry is responsive to detection of a trigger condition when executing a given debug instruction to exit the halted mode transparently to the debugger, and to take an exception in order to execute exception handler code comprising a sequence of instructions fetched from memory. On return from the exception, the processing circuitry then re-enters the halted mode and performs any additional processing required to complete execution of the given debug instruction. This provides a mechanism for allowing an apparatus to perform operations required by debug instructions in situations where the processing circuitry hardware is not able to natively perform those operations in response to the specified debug instruction.