The formal multiple scattering solution obtained previously for the scattering of a plane wave by an arbitrary configuration of parallel cylinders is applied to the finite grating of cylinders and to the reflection grating of semicylindrical bosses on a perfectly conducting infinite plane. A far field solution in closed form for spacing large compared to wavelength and radius is obtained on neglecting the ``end effects.'' It is found that for the transmission grating both polarizations may be markedly affected by multiple scattering; for the reflection grating, however, the effects are much more pronounced for polarization perpendicular to the elements. The case of radii≪&lgr; is investigated in detail; the &lgr;'s for which the effects are greatest are determined, and various intensity curves are presented. The bright and dark bands appearing on these curves (overlaying the usual continuous spectra observed with broadband radiation) are similar to the ``grating anomalies'' discovered by Wood in 1902. The present analysis provides a simple physical interpretation of their presence in terms of the magnitudes and phases of the various orders of scattering. Simple relations for their dependence on the parameters are derived which agree in general with experimental results. Both the electromagnetic and acoustic cases are treated.