This book reports on cutting-edge analysis of data from the Axion Dark Matter eXperiment (ADMX) which aims to detect axions in the dark matter halo of the galaxy. ADMX uses a high-Q microwave cavity in a strong magnetic field to capture photons generated by the decay of axions. The author's pioneering "high resolution" analysis of ADMX data was motivated by simulations suggesting that there are unvirialized streams of dark matter axions which pass the Earth's location. All have nearly the same velocity and hence the same relative kinetic energy. The frequencies of the photons from decay of these unvirialized particles would fall within a very narrow range of the RF spectrum, perhaps only a few tens of milliHertz in width. This is in contrast to the kHz width of the expected Maxwellian spectrum for the virialized component. The author extended the analysis to a set of spectral resolutions between 20 mHz and 1 Hz, demonstrating the improved signal-to-noise ratio obtained by his method. In addition, this reports the author's work on the design, construction, and testing of tunable cavities for the ADMX experiment, work which led to the manufacture of a four-cavity array to be used in the next year to extend the search to higher axion masses.