Abstract Larger telescopes are required by astronomers to see further into the depths of our Universe, in order to understand its origins and the origins of life. A solution is to replace conventional primary mirrors in space-telescopes by membrane reflectors that can be folded and fitted into a spacecraft before launch. Once deployed, the membrane would provide a large reflective surface necessary to obtain images of faraway celestial objects. Whilst the primary mirror of a space-telescope needs to have a very precise shape, membranes can have an uncertain shape for they are prone to wrinkling. For this reason, there is hesitation in using these structures for telescopes, as the technology needs to mature. In this thesis, regular polygonal membranes tensioned at their vertices are proposed as reflectors; their discrete nature necessitates a lighter support structure compared to circular membranes. Analysis has been made to assess the effect of the wrinkling in these polygonal membranes, where the primary focus has been on nominally flat membranes, alongside parabolic membranes. The area of reflectance of the membrane must be maximal and it is vital to mitigate wrinkles. In this thesis, the origins of the wrinkles are investigated and two novel stress analysis methods for polygonal membranes are developed initially. The first uses Airy stress functions: by superposing a number of these functions, and deriving from it a solution for a free body disk, the stress field of a polygon is approximated with a high degree of accuracy both in terms of distribution and magnitude. The second method uses the analogy of the governing equations of in-plane stresses and of curvatures in the shallow bending of plates. The solution produces the shape of stress fields accurately. The results from these two distinct methods have been validated using Finite Element Analysis (FEA) and, together, they conclude that compressive regions are located close to the corners of all regular polygons, no matter the number of sides. This information is used to develop a method to prevent compressive stresses in membranes; furthermore, the analogy is adapted to show that with a given amount of edge trimming, the tensioned membrane does not produce any compressive stresses and this is confirmed in practise. The minimum amount of trimming required is derived for any regular polygon. Experiments are performed on some of these polygons to compare the actual wrinkling regions with predictions obtained by analysis. It is shown that, although wrinkles do appear close to some corners in the experiment, not all corners wrinkle and the ones that do, only start wrinkling at relatively high tension forces. These experimental observations are not reproduced either by some previous theoretical work or FEA wrinkle predictions, where wrinkles have been shown to be present in all corners, even at low tension forces. A simple linear FEA has shown that the “connection” used to tension the membrane plays an important role in the wrinkling mechanism and the connections used in the experiments here are shown to produce the least amount of compressive stresses which can explain the results obtained. Generally speaking, it seems that the connections have a disproportionate effect on the wrinkle formation, not previously reported. It is also shown that physically trimming the edges results in a large increase in the unwrinkled area of the membrane. A methodology is developed to analyse the failure of a tension force which results in non- uniform external loads, leading to possible incursion of wrinkles over the bulk of membranes and which disrupts the reflective area substantially. It is essential to be able to vary the other loads through controllable tensions, and it is shown that, in doing so, wrinkles can be removed from the bulk of the membrane or that the reflective surface can be maximised. Finally, a study on the feasibility of “pressurising” the membranes into an out-of-plane shape has shown that electrostatic attraction is the best option given the free boundary conditions of the polygons. When pressurised, the polygonal membranes naturally reach a parabolic shape towards their centre, the extent of which varies greatly depending on a large number of parameters, including most particularly pre-tension, focal ratio and shape of the attractive surface. The study shows that specified accuracy requirements can be reached using this design in these regions. Generally, it is seen that polygonal membrane reflectors are good candidates to replace conventional primary mirrors. Even though some parts of the membranes are wrinkled or attain incorrect shapes, large areas can still be used for reflective purposes and recommendations on how to maximise these areas are given through the research performed in this thesis.