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This thesis concentrates on the development of a process-chain for the manufacture of the mirror segments for the European Extremely Large Telescope (E-ELT). This revolutionary scientific project with a 39.3m primary mirror will be the largest optical/near-infrared telescope in the world. The primary mirror design consists of 798 aspheric hexagonal segments, each 1.44 metres across-corners, but only 50mm thick. The manufacture of these aspheric segments poses many challenges. Edge mis-figure is regarded as one of the most difficult technical issues for segment production, impacting directly on the telescope’s science output, such as the detection of extrasolar terrestrial planets. The other challenge is how to speed up the process for the manufacture of almost 1000 segments in a reasonable time and cost. The 'Precessions' polishing technique is an advanced polishing method with high precision and efficiency. To achieve the specification of E-ELT’s segments, there are some key techniques that need to be researched, such as: edge control, removal of mid-spatial frequency errors and a massive improvement in process speed c.f. classical methods. The research work in this thesis contributed to the delivery of an effective process, including the characterisation of tool influence functions for the 'Precessions' technique; optimisation of the process parameters; edge control; diagnoses of edge asymmetry of hexagonal parts, and speeding-up the whole process. Two process-chains giving substantially different depths of removal have been developed and demonstrated on a hexagonal Zerodur part (200mm across-corners) with edge control, and the results meet the E-ELT specification. As a result, different top-level strategies for addressing the manufacture have been considered, leading to a preferred direction for the research.