An apparatus and method are described for handling memory access operations, and in particular for handling faults occurring during the processing of such memory access operations. The apparatus has processing circuitry for executing program instructions that include memory access instructions, and a memory interface for coupling the processing circuitry to a memory system. The processing circuitry is switchable between a synchronous fault handling mode and an asynchronous fault handling mode. When in the synchronous fault handling mode the processing circuitry applies a constraint on execution of the program instructions such that a fault resulting from a memory access operation processed by the memory system will be received by the memory interface before the processing circuitry has allowed program execution to proceed beyond a recovery point for the memory access instruction associated with the memory access operation. In contrast, when in the asynchronous fault handling mode, the processing circuitry removes that constraint. The processing circuitry is then arranged to switch between the synchronous fault handling mode and the asynchronous fault handling mode during execution of the program instructions, in dependence on a current context of the processing circuitry. This enables the apparatus to selectively take advantage of the higher performance associated with asynchronous reporting of faults, and the improved fault handling associated with the synchronous reporting of faults.