The systematic study of pseudodifferential operators draws lots of its motivation from partial differential equations and signal analysis. In turn, this study led to the introduction of the so-called isotropic (homogeneous and inhomogeneous)

Hormander classes of symbols. The adjective isotropic we use here points out that the spatial and frequency variables of the symbol obey the same scaling. However, in several examples (such as the heat operator) there exists another natural scaling (such as parabolic) and thus we fall in the realm of anisotropic symbols. The goal of this talk is to show the plausibility of a larger theory of anisotropic

pseudo-differential operators. Specifically, we define and study the boundedness properties of the (homogeneous and inhomogeneous) anisotropic generalizations of the H\"ormander classes above in the very general framework determined by an expansive dilation matrix A. This is joint work with M. Bownik (U Oregon).