The remarkable properties of polymers have made them ubiquitous in products, technologies, foods, medical devices, and more. These molecules have also presented many scientific challenges which have been met with an array of elegant theories over the past century. Today, polymer scientists turn their attention to ever more sophisticated macromolecules in search of new frontiers in polymer physics and new solutions to modern problems. Some of the most intriguing avenues of research center on mechanically-interlocked polymers (MIPs), which contain complex and permanent topological interactions in the form of mechanical bonds. Topological interactions, are crucial in determining the properties of polymeric materials with practical implications in many industrial processes. Owing to the unique and long-lived nature of the mechanical bond, MIP systems have been found to exhibit a host of unusual material properties including excellent toughness and resilience, which has made them promising candidates for several technological applications. Despite the promise of MIPs, scientists and engineers must still contend with many challenges in applying these polymers. In addition to the well-known synthetic difficulties associated MIPs, researchers are also faced with a dearth of knowledge concerning the properties of MIP systems, both in the basic phenomenology and the underlying physics. This is especially apparent in the case of poly[n]catenanes - polymers which consist entirely of interlocking ring molecules. Such systems have been targeted by chemists for decades and were recently synthesized for the first time, but their properties remain largely unexplored. This dissertation alleviates some of these difficulties by using molecular dynamics (MD) simulations to examine the physics of poly[n]catenanes. To begin, the force-extension behavior of the polymers is studied with atomistic resolution and the results are found to agree well with experiments. To access longer length/time scales, coarse-grained models are then used to study poly[n]catenane structure and dynamics in good solvent conditions. It is found that topological interactions greatly slow the dynamics at short and intermediate length scales in a manner similar to entanglement in linear polymer systems. Next, the effect of hydrodynamic interactions in catenated polymers is considered using both theory and simulation and we find that the symmetries of ring molecules greatly simplify the situation and the effects of topology and hydrodynamics may be considered roughly independently. Large scale MD simulations are then used to study poly[n]catenanes in the melt. It is found that the mechanical bonds cause large reductions in the pressure and lead to complex polymer conformations that resist interchain entanglement. Despite the lack of such entanglements, the dynamics of poly[n]catenane melts are extremely rich, although some qualitative features can be explained by a simply, analytically solvable model. Most strikingly, the viscosity exhibits a non-monotonic dependence on the ring size, with larger rings (i.e. bigger polymers) leading to lower viscosities up to a critical ring size, above which the trend reverses and the viscosity increases with ring size. To conclude, implications for chemical synthesis and materials design are discussed and directions for future research are proposed.