The formation of polymer ring structures leads to new physical effects that arise from topological restrictions. These restrictions consist of the fact that the topological state of a ring polymer molecule cannot vary. Interest in this type of object has increased sharply since the proof that in most cases the molecules of DNA function in the cell in a closed circular form. This review presents the necessary information on the physical properties of circular DNAs. The mathematical basis of the theory of knots and linkages are briefly presented, together with the theory of ribbons. This apparatus is employed to obtain quantitative data on the behavior of closed polymer chains by Monte Carlo calculations on a computer. The results of the calculations are compared with the experimental data for circular DNAs. This comparison has made it possible to obtain valuable information on the properties of the double helix, such as its torsional rigidity. The biological role of the topological restrictions in DNA is discussed, as well as the significance of the recently discovered enzymes, topoisomerases, which alter the topological state of circular DNAs.