Cyclic ozone (O3) has not been isolated so far, despite its computed kinetic persistence. Possibilities of "trapping" this molecule (or the valence-isoelectronic cyclic thiozone, S3) in transition metal complexes are investigated in this paper. Candidates were constructed, first using the 18-electron rule as a guide and then optimizing the structures with the DFT-B3LYP method. A variety of structures result: oxo-peroxo species, di-sigma- and pi-bonded open ozone complexes, some eta1 and eta2 cyclic ozone complexes, and a few bona fide eta3 cyclic O3 and S3 complexes. MLn fragments suitable for complex formation would need to contain very strong pi-acceptor ligands. Nitrosyl ligands were chosen to minimize an energy mismatch between the O3 donor orbitals and the MLn acceptor orbitals. On this basis, the existence of the complexes [S3W(NO)3]3+, [O3M(NO)3]3+ (M = Cr, Mo, W, Fe, Ru, Os), and [S3W(NO)2(CO)]2+ containing cyclic O3 and S3 is suggested. In another approach, facing up to the oxidizing power of O3, potential systems were built from late transition metals in high oxidation states, and also d0 early transition metal centers.