The most important structural application of cellular solids is in light-weight sandwich structures where they are used as a core material. A sandwich is a laminated composite consisting of two dense high-performing skins bonded together with a light core. The sandwich design is efficient for structures working under bending loads, and is common in wind turbine blades and boat hulls.Although usually low-performing, core materials have to meet sufficient mechanical requirements to allow transfer of the acting load from one skin to another without failure. The most commonly used cores today are balsa wood and polyvinyl chloride (PVC) foams. Despite their good mechanical properties, there are certain disadvantages associated with their use in practice. Balsa is a hydrophilic material, which is of concern in marine applications. PVC foams have an outgassing problem, which can be a reason for delamination in sandwich structures. Both balsa wood and PVC foams are expensive. In order to promote development of new core materials, a good understanding of the structure-property relations of cellular solids with closed-cell structures is essential.The goal of this PhD research was to systematically study the behaviour of popular core materials, such as balsa wood, PVC and styrene acrylonitrile (SAN) foams, as well as nano-reinforced polyurethane (PU) and polypropylene (PP) foams that are currently still under development. The study was carried out using experimental methods supported by modelling investigations. The experimental methodology included micro-CT image analysis of the cellular structure, compression tests on two structural levels and fracture toughness evaluation. Some of the tests are not standardised and were designed specifically for this study. The established structure-property relation helped identify strategies that can be implemented to achieve, for a given density, higher Youngs moduli in cellular materials. These strategies included increasing stiffness of the solid polymer and adjusting it with the foam density; creating an anisotropic cell structure; minimising bending deformation of cell walls. Balsa wood was found to have superior elastic properties, explained by its sophisticated multi-level structure where all aforementioned strategies were well implemented. Among foams, PVC and SAN foams were the best alternatives to balsa wood. Their microstructures and the polymer properties were found to be tuned with foam density.While nano-reinforced PP and PU foams underperformed in comparison with commercially available foams, they showed potential for further improvement. Nanoclays were found to help with nucleation of additional bubbles and production of microcellular structures. The average cell size of the nanoclay-reinforced foams decreased in comparison with the pure foams, and it also resulted in higher stiffness of the foams. The influence of nanofillers on foam mechanical properties was twofold. For brittle PU foams, nanoclays acted as stress concentrators in cell walls, leading to a decrease of the mechanical properties. By contrast, ductile PP foams showed improved performance after addition of both nanoclays and carbon nanofibres.