One of the most significant occurrences in diffuser design, if not the most important event, was the invention of the phase grating diffuser by Schroeder.1,2 Apart from very simple constructions, previous diffusers had not dispersed sound in a predictable manner. The Schroeder diffuser offered the possibility of producing optimum diffusion and also required only a small number of simple design equations. D'Antonio and Konnert3 presented one of the most readable reviews examining the far field diffraction theory underpinning Schroeder's number theoretic surfaces; they experimentally measured performance and described applications in critical listening environments. Most crucially, they commercialized Schroeder diffusers and so made them widely available. During the four decades since the diffusers were invented, there have been many new developments, which are documented in this chapter. Some of this chapter was featured as a review article in the journal Building Acoustics.4