As far as we know, only a few studies have been, up to now, devoted to anisotropic gratings. We only have heard of the paper by our Japanese colleagues of Osaka Prefecture University.1 We are also working in this field, in view of potential applications. Although some partial and unpublished results have been obtained some years ago in the framework of a contract with a private company (P.A. Technology, Cambridge), we think that we have to face a difficult problem which is far from being wholly resolved. We would like to report briefly on our theoretical work in order to inform "practical people" and, should the occasion arise, to initiate new collaborations. Throughout the paper we use a rectangular coordinate system Oxyz and we denote by ex, ey, ez the unit vectors (4, axes Ox, 0y, Oz. We deal with time-harmonic fields represented by complex vectors t and H, using a time dependence in exp(-iwt). The permeability is equal to μ0 everywhere and the relative permittivity is described by a 3 x 3 matrix  : xx xy xz yxyy yz zx zy zz We denote by co the vacuum permittiviV, an4 we put ko = wicopo = 27/X0. We say that the fields are TE (or TM) polarized when E (or H) is parallel to Oz.