In Castaing’s classic Ph.D. dissertation he described how the limiting value of x-ray spatial resolution for x-ray microanalysis, of about 1 μm, was not imposed by the diameter of the electron beam, but by the size of the region excited inside the specimen. Fifty years later this limit still applies to the majority of measurement made in EMAs and SEMs, even though there is often a need to analyze much finer structures. When high resolution chemical analysis is required, it is generally necessary to prepare thin sections and examine them in an analytical electron microscope where the maximum diameter of the excited volume may be as small as a few nanometers. Since it is not always possible or practical, it is important to determine just what is the best spatial resolution attainable for the examination of polished or “as received” samples with an EMA or SEM and how to achieve it experimentally.