The structural vibration damping behaviour of plates and beams can be improved by the application of viscoelastic passive damping materials. Unconstrained layer damping treatments applied to metal plate systems were studied experimentally. Design and modelling of novel fibre reinforced constrained layer damping materials was performed, and implementation of these composite damping materials into laminated composite sandwich constructions commonly used as structural elements within large composite marine vessels was explored. These studies established effective methods for examining, designing and applying damping materials to metal and composite marine structures. Two test fixtures were designed and constructed to facilitate testing of viscoelastic material damping properties to ISO 6721-3 and ASTM E756. Values of material damping made in accordance with ASTM E756 over a range of temperatures were compared to values produced by a Dynamic Mechanical Analyser (DMA). Glass transition temperatures and peak damping values were found to agree well, although results deviated significantly at temperatures above the glass transition temperature. The relative influence of damping layer thickness, ambient temperature, edge conditions, plate dimensions and substrate material on the system damping performance of metal plates treated with an unconstrained viscoelastic layer was investigated experimentally. This investigation found that substrate material had the greatest influence on system damping performance, followed by damping layer thickness and plate size. Plate edge conditions were found to have little influence on the measured system damping performance. These results were dependent on the values of each variable used in the study. Modal damping behaviour of a novel fibre reinforced composite constrained layer damping material was investigated using finite element analysis and experimental methods. The material consisted of two carbon fibre reinforced polymer (CFRP) layers surrounding a viscoelastic core. Opposing complex sinusoidal fibre patterns in the CFRP face sheets were used to achieve stress-coupling by way of orthotropic anisotopy about the core. A finite element model was developed in MATLAB to determine the modal damping, displacement, stress, and strain behaviour of these complex patterned fibre constrained layer damping (CPF-CLD) materials. This model was validated using experimental results produced by modal damping measurements on CPF-CLD beam test specimens. Studies of multiple fibre pattern arrangements found that fibre pattern properties and the resulting localised material property distributions influenced modal damping performance. Inclusion of CPF-CLD materials in laminated composite sandwich geometries commonly used in marine hull and bulkhead constructions was studied experimentally. Composite sandwich beam test specimens were fabricated using materials and techniques frequently used in industry. It was found that the greatest increases in modal damping performance were achieved when the CPF-CLD materials were applied to bulkhead geometries, and were inserted within the sandwich structure, rather than being attached to the surface.