Following the introduction of quantifiable number theoretic, reflection phase grating diffusing surfaces by Schroeder [J. Acoust. Soc. Am. 57, 149–151 (1975)], several approaches have been used to measure and characterize them. In the early 1980s, time delay spectrometry, developed by Heyser, was used to measure the temporal, spatial, and spectral responses of diffusing surfaces. Once maximum length sequences were introduced as an excitation signal, they were adopted in the late 1980s. These explorations led to the creation of a measurement standard for the diffusion coefficient, which is the circular autocorrelation of the third octave polar responses. The limitations of periodic number theoretic surfaces, namely, bandwidth, grating lobes, and flat plate effects, were addressed by computer optimization, which combined the power of boundary element and multi‐dimensional minimization techniques, and modulation. Both one‐dimensional (phase variation in one direction) and two‐dimensional (phase variation in two orthogonal directions) surfaces were characterized. The data collection and reduction procedures will be reviewed with several examples of diffusing surfaces. A summary of this research can be found in the work of Cox and D’Antonio [Acoustic Absorbers and Diffusers: Theory, Design and Application (Spon, 2004)].