Viroids are single-stranded circular RNA molecules of 240 to 400 nucleotides which are pathogens of certain higher plants and replicate autonomously in the host cell. Virusoids are similar to viroids in respect to size and circularity but replicate only as genomic part of a plant virus. Their structure and structural transitions have been investigated by thermo-dynamic, kinetic and hydrodynamic methods. The special features of the sequences of these RNAs, which are the basis for their secondary structures and structural flexibility, are investigated with theoretical methods. A set of thermodynamic parameters for helix growth and loop formation is selected from the literature to calculate secondary structures and structural transitions of single-stranded RNAs. Appropriate modifications of the chosen parameter set are discussed. For calculations we used either Tinoco-plots and the model of "cooperative helices" or the Zuker-program based on the exact algorithm of Nussinov et al, or both. Calculations were done for viroids and virusoids. As both are single-stranded, circular RNAs we had to modify the Zuker-program as described in the appendix. Calculations are done for different viroids, i.e. potato spindle tuber, citrus exocortis, chrysanthemum stunt, coconut cadang-cadang, and avocado sunblotch, and for two virusoids, i.e. the circular RNAs of Solanum nodiflorum mottle virus, and velvet tobacco mottle virus. For viroids the calculations confirm our earlier theoretical and experimental results about the extended native structure and the highly cooperative transition into a branched structure. Virusoids show less base pairing, branching in the native secondary structure, and only low cooperativity during denaturation. They resemble more closely the properties of random sequences with length, G:C content, and circularity as in viroids but statistical sequences. The comparison of viroids, virusoids, and circular RNA or random sequences confirms the uniqueness of viroid structure.