A method of operating a processing unit of a computer system, by issuing an instruction having an explicit prefetch request directly from an instruction sequence unit to a prefetch unit of the processing unit. The invention applies to values that are either operand data or instructions. In a preferred embodiment, two prefetch units are used, the first prefetch unit being hardware independent and dynamically monitoring one or more active streams associated with operations carried out by a core of the processing unit, and the second prefetch unit being aware of the lower level storage subsystem and sending with the prefetch request an indication that a prefetch value is to be loaded into a lower level cache of the processing unit. The invention may advantageously associate each prefetch request with a stream ID of an associated processor stream, or a processor ID of the requesting processing unit (the latter feature is particularly useful for caches which are shared by a processing unit cluster).