Plasma opacity calculations play an important role in solar modelling and many plasma physics and inertial confinement fusion experiments. This thesis is focussed on the fast calculation of opacity from first principles. The existing average atom (AA) opacity code IMP [1] is used alongside experimental data and detailed atomic physics to develop new models; the results show that simple models can give an excellent description of plasma spectra for a large range of conditions. The results are significant for the development of fast opacity codes which necessarily use the AA approach. The application of fast models to very large scale calculations is considered and an efficient approach to these developed; this allows the fast description of experimental data that would not have otherwise been possible [2]. Analysis of this data then allows the accuracy of the IMP model to be further discussed. The atomic model is also considered, and an improved approach implemented. These improvements makes little difference to the description of experiment provided electron exchange is included. The range of applicability of the IMP model is then extended to higher density by adding a fast description of line broadening by electrons. This gives an excellent agreement with both experiment and more advanced opacity codes. The treatment of atomic term structure can represent a significant portion of code runtime. A good compromise between detail and efficiency is the unresolved transition array (UTA) formulation; a consistent theory of UTAs is developed, and various models introduced. The accuracy of these is systematically tested. It is found that within the validity range of the UTA approach, a good description of the opacity can be gained using a simple model provided that the linewidth is correct. Various simplified calculations of this width are tested, and found to be inaccurate [3].