Collective P-and T-odd moments produced by parity and time invariance violating forces in reflection asymmetric nuclei are considered. The enhanced collective Schiff, electric dipole and octupole moments appear due to the mixing of rotational levels of opposite parity. These moments can exceed singleparticle moments by more than two orders of magnitude. The enhancement is due to the collective nature of the intrinsic moments and the small energy separation between members of parity doublets. In turn these nuclear moments induce enhanced T-and P-odd effects in atoms and molecules. First a simple estimate is given and then a detailed theoretical treatment of the collective T-, P-odd electric moments in reflection asymmetric, odd-mass nuclei is presented and various corrections evaluated. Calculations are performed for octupole deformed long-lived odd-mass isotopes of Rn, Fr, Ra, Ac and Pa and the corresponding atoms. Experiments with such atoms may improve substantially the limits on time reversal violation.