Ring polymers offer a richness of behaviours that are of broad interest and have deep consequences in many fields of Science. In this Thesis I investigate some general and universal properties, i.e. independent of the chemical nature of the polymers, emerging from systems made of a collection of rings. These will be studied by using methods of equilibrium and non-equilibrium Statistical Mechanics together with Molecular Dynamics and Monte Carlo simulations of coarse-grained models for the systems under investigation. Within these frameworks, important questions regarding the macroscopic behaviour of ring-shaped polymers have yet to find a satisfactory answer. The work presented in this Thesis finds its principal motivations in problems arising in Material Science, the so called \melt" of rings, and in Biology, such as the organisation of mitochondrial DNA in some organisms and the mechanisms governing the electrophoretic separation of DNA samples in gels. There are several theoretical challenges in these fields which represent state-of-the-art scientific research and whose partial answers are provided in the work presented in this Thesis. One of the major achievements of the work presented is the general understanding of the role played by topological properties, i.e. those invariant under smooth deformations of the polymer contour, on the macroscopic behaviour of the investigated systems. Finally, the conclusions drawn from the presented work can have important scientific consequences as they may ultimately lead to a more complete understanding of complicated issues in Biology and to the design of next-generation soft materials.