The Fraunhofer patterns of blazed gratings are derived on the basis of a scalar theory which includes the non-linear dependence of the obliquity factor and the phase modulation on the spatial frequencies defining the positions of the source and of the observer. The solution based on the usual 'linear communications' theory is compared with one based on the more general non-linear theory; it is shown that the former is meaningful only in the neighborhood of the blaze wavelength. The behavior of blazed gratings is examined in the light of non-linear theory in the region away from the blaze wavelength. It is shown that the envelope function describing the amplitude distribution due to a single groove depends on the single parameter defining half the phase difference between the two edges of a single diffracting facet. It is also shown that certain wavelengths are missing from the zero order and that 'dark' lines exist into which no light of any order is transmitted. A useful maximum for the aspect ratio is derived. The Littrow and spectrograph configurations are examined in some detail.