Numerically controlled optical finishing machine tools equipped for selective material removal to produce non-spherical optical surfaces having all of the following characteristics: (a) Finishing the form to less (better) than 1,0 μm; (b) Finishing to a roughness less (better) than 100 nm rms; (c) Four or more axes which can be coordinated simultaneously for ‘contouring control’; and (d) Using any of the following processes: 1. ‘Magnetorheological finishing (MRF)’; 2. ‘Electrorheological finishing (ERF)’; 3. ‘Energetic particle beam finishing’; 4. ‘Inflatable membrane tool finishing’; or 5. ‘Fluid jet finishing’. Technical notes : For the purposes of the items above: 1. ‘MRF’ is a material removal process using an abrasive magnetic fluid whose viscosity is controlled by a magnetic field. 2. ‘ERF’ is a removal process using an abrasive fluid whose viscosity is controlled by an electric field. 3. ‘Energetic particle beam finishing’ uses Reactive Atom Plasmas (RAP) or ion beams to selectively remove material. 4. ‘Inflatable membrane tool finishing’ is a process that uses a pressurized membrane that deforms to contact the workpiece over a small area 5. ‘Fluid jet finishing’ makes use of a fluid stream for material removal