We develop a mathematical foundation for "operator quantum error correction". This is a new paradigm for the error correction of quantum operations that incorporates the known techniques -- i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method -- as special cases, and relies on a generalized notion of noiseless subsystems that is not restricted to the commutant of the interaction algebra. We establish conditions on the noise operators for a given quantum operation that characterize both correctability and the existence of generalized noiseless subsystems. The condition from the standard model is shown to be a prerequisite for any of the known forms of error correction. We present a new class of quantum channels and discuss subsystems that are immune to noise up to unitary conjugation.