In this paper we consider linear ill-posed problemswhere instead of y noisy data yδare available withandis a linear operator between Hilbert spaces X and Y. Assuming the general source conditionwith appropriate functions φ we study following questions:(i) which (best possible) accuracy can be obtained for identifying x fromunder the assumptions(ii) are there special regularization methods which guarantee this best possible accuracy, i.e., which are optimal on the set Mδ,E? Concerning question (i) we prove that under certain conditions there holds inf supwithwhere the ‘inf’ is taken over all methodsand the 'sup' is taken over all.andConcerning question (ii) we prove the optimality of a general class of regularization methods and specify our general optimality results to Tikhonov type methods and to spectral methods. Heat equation problems backward in time which are characterized by different functions φ(λ) serve as model examples.