The resistance of glass and carbon fibre reinforced polymer (GFRP and CFRP) sandwich panels and laminate tubes to blast in air and underwater environments have been studied. Explosive charges of 0.64-100 kg TNT equivalent were used during these studies. Procedures for monitoring the structural response of such materials during blast events have been devised. High-speed photography was employed during the air-blast loading of GFRP and CFRP sandwich panels, in conjunction with digital image correlation (DIC), to monitor the deformation of these structures under shock loading. Failure mechanisms have been revealed using DIC and confirmed in post-test sectioning. Strain gauges were used to monitor the structural response of similar sandwich materials and GFRP tubular laminates during underwater shocks. The effect of the supporting/backing medium (air or water) of the target facing the shock has been identified during these studies. Mechanisms of failure have been established such as core crushing, skin/core cracking, delamination and fibre breakage. Strain gauge data supported the mechanisms for such damage. A transition in behaviour was observed in the sandwich panels when subject to an underwater blast as opposed to an air-blast load. Damage mechanisms notably shifted from distributed core shear failure originating from regions of high shear in air blast to global core crushing in underwater blast. These studies were part of a research programme sponsored by the Office of Naval Research (ONR) investigating blast loading of composite naval structures. The full-scale experimental results presented in this thesis will aid and assist in the development of analytical and computational models. Furthermore, this work highlights the importance of support and boundary conditions with regards to blast resistant design. These outcomes were analysed further in finite element simulations of both air and underwater blast conditions, where boundary stiffness and support conditions were, as expected, shown to strongly influence structural response and deformation of the target.