Polymeric foams are extensively used as the core materials in sandwich structures and the core material is typically bonded between relatively thin fibre-composite skins. Such sandwich structures are widely used in the aerospace, marine and wind-energy industries. In the present work, various sandwich structures have been manufactured using glass-fibre-reinforced polymer (GFRP) skins with three layers of poly(vinyl chloride) foam to form the core, with the densities of the foam layers ranging from 60 to 100 kg/m3. This study has investigated the effects on the quasi-static flexural and high-velocity impact properties of the sandwich structures of: (a) the density of the polymeric-foam core used and (b) grading the density of the foam core through its thickness. The digital image correlation technique has been employed to quantitatively measure the values of the deformation, strain and onset of damage. Under quasi-static three-point and four-point bend flexural loading, the use of a low-density layer in a graded-density configuration reduced the likelihood of failure of the sandwich structure by a sudden force drop, when compared with the core configuration using a uniform (i.e. homogenous) density layer. The high-velocity impact tests were performed on the sandwich structures using a gas-gun facility with a compliant, high-density polyethylene projectile. From these impact experiments, the graded-density foam core with the relatively low-density layer located immediately behind the front (i.e. impacted) GFRP skin was found to absorb more impact energy and possess an increased penetration resistance than a homogeneous core structure.