What can be expected from the further study of CP and T symmetry violation and CPT invariance tests

Different CP and T violation models are discussed in view of using their particular features to experimentally ascertain which of these models are realized in nature. For example, recording the electric dipole moments of the neutron and electron close to their experimental limits or only one or two orders of magnitude smaller would be evidence for the presence of CP violation sources other than the Kobayashi - Maskawa phase in the Standard Model. As for CPT invariance, which predicts, in particular, the equality of the particle and antiparticle masses, the ratio of the difference between the K0 and K-bar0 masses to their sum being below 10-18 is considered to be the best test of CPT symmetry. However, an extremely small value of this ratio does not necessarily imply extremely small parameters of the CPT violation in {K0, k-bar0}-system decays. The existing data only indicate that these parameters must not exceed 30% of the known CP violation parameter η+-. Other experiments for high-precision testing of CPT invariance are discussed. (reviews of topical problems)