Polymers are intriguing physical systems whose complex properties are at the heart of how viscoelastic substances, materials which under strain manifest a behavior which is intermediate between a liquid and a solid, work. Understanding the properties of these materials is the main goal of the theoretical and computational tools of Polymer Physics. A particularly important, yet not fully understood, class of polymer materials is represented by concentrated solutions and melts of unknotted and unconcatenated ring polymers: in fact, at odds with the more familiar case of linear polymers which tend to become highly mixed and mutually penetrating, the presence of mutual avoidance and topological constraints (entanglements) between ring polymers force these chains to remain \u201cterritorial\u201d, i.e. each chain is virtually unmixed from the rest of the others. Because of this feature, solutions of ring polymers display unique material properties, in particular single chains tend to crumple into highly branched conformations and feature marked corrugated surfaces. Recently, it has been suggested that the spatial configurations of ring polymers in solution can be used as model systems for the organization of chromosome conformations during interphase, i.e. inside the nuclei of eukaryotic cells. This surprising analogy is built upon the claim that chromosomes undergo slow relaxation inside the nucleus which results in the spontaneous formation of so-called territories, regions of the nucleus which have a profound impact on crucial cellular functions such as gene expression and gene regulation. In this Thesis, we explore the analogy between ring polymers in solution, their large-scale crumpled 3d structure and interphase chromosomes by employing a combination of the theory of polymer solutions and numerical simulations. In more detail, we investigate primarily the following aspects: (a) the formation of ordered domains on a simple Ising-like toy model for crumpled polymers; (b) The analysis of the viscoelastic properties of model chromosome conformations whose stochastic motion is restricted by the presence of external constraints; (c) The discussion of the viscoelastic properties of solutions of active vs. non- active rings, where \u201dactive\u201d means that polymers are driven out-of-equilibrium by pumping energy inside the system
