Usual way to characterize the quality of the different a posteriori parameter choices is to prove their order-optimality on the different sets of solutions. In this paper we introduce the property of the quasioptimality to characterize the quality of the rule of the a posteriori regularization parameter choice for concrete problem Au = f. If the method P with a priori parameter choice is order optimal on the set of solutions M, then it is order-optimal with quasioptimal a posteriori parameter choice on the set M also. We consider two concepts of the quasioptimality and discuss the quasioptimality of different well-known rules for the a posteriori parameter choice as discrepancy principle, the modification of the discrepancy principle, Lepskii principle and monotone error rule. We consider also the family of weakly quasioptimal rules for a posteriori parameter choice.