This proposed project explores fundaments and principles for surface measurement and characterization for ultra/nano-precision non-Euclidean geometry and deterministic surfaces, which are vital for making possible key areas of 21st century science - pure and applied, engineering and bio-engineering. The research will explore an original integrated measurement and characterization system with two major aspects: (1) Characterization: to develop radical new thinking as to what are the fundamental building blocks of a texture-characterization system and apply that thinking to non-Euclidean and deterministic surfaces. It will explore necessary and sufficient mathematical operations and principles, surface decomposition models, distortion-free representation of texture etc. (2) Measurement: to investigate principles and enabling optical methods to on-line/in-line measurement for ultra/nano-precision non-Euclidean geometry and deterministic surfaces. The fruits of this research project will significantly facilitate surface-manufacturing control and functional performance of surfaces applied in 21st Century Science and Engineering over a wide set of sectors. Examples are surfaces used in optics and target shells in high-power laser-energy systems, optics in new earth/space-based large telescopes (e.g. the 42 m E-ELT telescope), interfaces in fluid-dynamics (energy-efficient jet engines, aircraft fuselages and wings), long-life human-joint implants, microelectronics and MEMS/NEMS devices in nanotechnology applications. The capability to perform surface quantitative measurements and characterization on the above key components does not exist today. This confronts the state-of-the-art in surface-measurement science with regard to new surface characteristics (structured or patterned surfaces), extremity of size (1 m - 2 m), ultra precision (1 in 10^9), quality, complexity of shape (non-Euclidean geometry), or combinations of these aspects.