In this Thesis, the suitability of 37 actuator Micro-machined membrane deformable mirrors (MMDM) for adaptive optics at cryogenic temperatures is evaluated. A method is described to safely cool MMDMs to 78K. The initial surface figures of three MMDMs are examined using a Zygo phase shifting interferometer to evaluate the effects of both evacuating and cooling the MMDM. The cooling process increases the aberrations inherent in the MMDMs' membranes. One of the MMDMs evaluated is found to have intrinsic aberrations sufficiently small to leave substantial dynamic range free for wavefront correction; the available range is determined by the beam diameter being corrected. The influence functions of the actuators of the MMDMs are then examined; the cooling process leaves the form of the influence functions unchanged, but reduces the magnitude by 20%.The dynamic response is evaluated for MMDMs up to a driving frequency of 500Hz. Evacuation, which removes air damping, has no effect on the dynamic response. Cooling the MMDMs shows a high frequency ringing at a frequency of 2kHz, irrespective of MMDM driving frequency. This effect is very small for the MMDM with the superior initial surface figure, which can be considered to be suitable for operation in a cryogenic adaptive optics system. This was the first demonstration of an adaptive optical element functioning at cryogenic temperatures. A simulation is developed to demonstrate that an adaptive optics system could be constructed utilising a MMDM and a curvature sensor that corrects wavefronts without a wavefront reconstruction step. This system gives good correction for defocus, coma and spherical aberration, with reasonable correction for astigmatism. Potential avenues for future exploration are discussed, notably the prospect of optimising the design of MMDMs for cryogenic operation.