This thesis is primarily aimed at carrying out analysis of Energy Bags, reinforced fabric bags used for subsea compressed air energy storage. Subsea compressed air energy storage is a completely new method of large-scale energy storage designed to be integrated with direct-compression offshore wind turbines and wave energy converters. Energy Bags are impermeable bags anchored to the seabed at significant depths (e.g. 500m) in which high pressure air, compressed by specially designed wind turbines and wave energy converters, is stored at pressures roughly equal to the hydrostatic pressure of the surrounding water. Energy Bags do not need to be particularly strong because most of the reaction to the pressure load is provided by the surrounding water, and high energy densities are available at such depths as 500m. This thesis investigates the deformed shapes of Energy Bags and studies optimal designs. Three analysis methods are developed which vary in their complexity, ease of use, and accuracy. First, a system of coupled ordinary differential equations (ODEs) is derived which describes the deformed shape of an axisymmetric Energy Bag. This model is later used in an optimisation study to find the shapes of bag which minimise the cost of materials (reinforcement, fabric, and ballast) per unit of energy stored. Circumferential reinforcement, hanging masses from the inside of the bag (which it was hoped would lower the total cost) and fill level are all included as variables in the optimisation, and it is found that for reasonable materials costs an Energy Bag could cost less than £10,000/MWh when anchored at 500m. This compares favourably with all other methods of large-scale energy storage. However, the bags used in the optimisation study have wide bases, which will require sealing against the seabed (unless water is to be allowed into the bags). Problems are encountered when trying to use the ODE method to find the shapes of partially inflated bags, and it is generally not very easy to use. Next, we carry out finite element analysis (FEA) of an axisymmetric Energy Bag using cable elements. This is much more user-friendly and flexible than the ODE method. Partially inflated bag shapes are found, and pressure-volume curves are presented which show the almost isobaric performance of an Energy Bag. It is found that material mass limits the extent to which the bag can be deflated before it becomes unstable. The axisymmetric FEA is used to study bags with much more realistic circumferential reinforcement than the ODE method, and we also look at bags with an unsealed base, which allow water in through the base as they deflate. A three-dimensional FEA tool is presented which models an Energy Bag as a cable-reinforced membrane using cable and membrane elements, and special measures had to be taken to deal with wrinkling. We assume that the bag is rotationally symmetric, comprising a number of symmetric lobes. The 3D FEA is used to find the stress distribution in the membrane of the bag, however a converged solution cannot always be found. It is not certain why this is the case but it is anticipated that it is because deformed bags are not always rotationally symmetric. The 3D FEA could also be used to model other membrane structures such as balloons, parachutes, roofs and sails, as well as nets. The standard cutting patterns for lobes in lobed balloons are analysed, and a new cutting pattern known as the Constant Tension lobe is generated. This is an extension of the Constant Radius lobe and takes into account the pressure gradient found in both air and water, minimising waste material. The Constant Tension lobe is particularly appropriate for Energy Bags because of the large pressure gradient in water. The Ultra High Performance Vessel architecture is also presented, upon which the design of the prototype Energy Bags is based. The fabric structure of an Ultra High Performance Vessel comprises only two sheets of fabric (rather than many separate lobes welded together), and tendon shortening and “bellows” serve to ensure that there is no meridional stress in the fabric. An analytical optimisation is used to show that the zero pressure bag that minimises cost of materials per unit of energy stored has equal costs of reinforcement and membrane. The axisymmetric FEA is also used to find the optimum bag size and maximum fill level for a bag which comes down to a single point at the base (as opposed to a wide base bag). Finally, testing of two 1.8m diameter superpressure Energy Bags has been commenced during the course of this work, and the prototypes and test rig are documented in this thesis. The prototypes were manufactured for us by Thin Red Line Aerospace Ltd., a Canadian manufacturer of deployable fabric structures for use in space. They are being cycled back-to-back in order to prove the concept, assess the performance of an Energy Bag over time, and identify any problems that need to be addressed. One of the bags had a few small leaks from the moment it was first inflated, but the other has remained airtight to date. It was found that if an Energy Bag is to be airtight, special attention must be paid to the welds at the seams and the sealing around the airline fittings.