Early depth test stages of a graphics processing pipeline broadcast information about fragments having conditional discard tests associated with them and that pass those early depth tests to other stages in the pipeline. The other stages in the pipeline use the early depth test pass information to determine if the processing of any fragments that they are currently processing can be slowed down. If a fragment that triggered the slowing down of the processing of fragments that are already in the pipeline passes all the conditional discard tests it is to be subjected to, a signal that triggers stopping of the processing of the fragments whose processing it had previously slowed down is sent. If the fragment fails a conditional discard test it is to be subjected to, a signal triggering the reviving of the processing of the fragments whose processing it had previously slowed down is sent.