After a brief history of near-field antenna measurements with and without probe correction, the theory of near-field antenna measurements is outlined beginning with ideal probes scanning on arbitrary surfaces and ending with arbitrary probes scanning on planar, cylindrical, and spherical surfaces. Probe correction is introduced for all three measurement geometries as a slight modification to the ideal probe expressions. Sampling theorems are applied to determine the required data-point spacing, and efficient computational methods along with their computer run times are discussed. The major sources of experimental error defining the accuracy of typical planar near-field measurement facilities are reviewed, and present limitations of planar, cylindrical, and spherical near-field scanning are identified.