The purpose of this research is to develop fundamentals of compliant mechanisms in deployable space systems. The scope was limited to creating methods for thick origami, developing compliant deployable solar arrays, and developing methods for stowing and deploying the arrays. The research on actuation methods was focused on a one-time deployment of the array. Concepts for both passive and active actuation were considered. The primary objective of this work was to develop approaches to accommodate thickness in origami-based deployable arrays with a high ratio of deployed-to-stowed diameter. The HanaFlex design was derived from the origami flasher model and is developed as a deployable solar array for large arrays (150 kW or greater) and CubeSat arrays (60 W). The origami folding concept enables compact stowage of the array, which would be deployed from a hexagonal prism into a flat array with about a 10-times increase in deployed diameter as compared to stowed diameter. The work on the origami pattern for the solar array was also applied to the folding of 80-100 m2 solar sails for two NASA CubeSat missions, NEA-Scout and Lunar Flashlight. The CubeSat program is a promising avenue to put the solar array or solar sails into space for testing and proving their functionality. The deployable array concept is easily scalable, although application to CubeSats changes some of the design constraints. The thickness-to-diameter ratio is larger, making the issues of thickness more pronounced. Methods of actuation are also limited on CubeSats because of the rigorous size and weight constraints. This dissertation also includes the development of a compact, self-deploying array based on a tapered map fold design. The tapered map fold was modified by applying an elastic membrane to one side of the array and adequately spacing the panels adjacent to valley folds. Through this approach, the array can be folded into a fully dense stowed volume. Potential applications for the array include a collapsible solar array for military or backpacking applications. Additional compliant mechanism design was done in support of the HanaFlex array. This included a serpentine flexure to attach the array to the perimeter truss for deployment, and a bistable mechanism that may be used in the deployment of the array or sail.