Abstract: The talk deals with scales of function spaces that are defined in terms of the decay behaviour of scalar products with elements of suitably defined systems of functions in ${\rm L}^2(\mathbb{R}^n)$. Examples of such scales are the Besov spaces (both the isotropic and anisotropic, as well as the homogeneous and inhomogeneous versions), ($\alpha$-)modulation spaces, shearlet coorbit spaces, or curvelet smoothness spaces. All named examples are in fact special cases of so-called {\em decomposition spaces}. These spaces have a Fourier-analytic definition, based on the choice of a suitable covering of the frequencies. The talk deals with the fundamental question when different coverings result in the same scale of function spaces. It turns out that this is equivalent to the quasi-isometry of certain metrics induced by the underlying coverings. As an illustration of the abstract theory, I present the classification of anisotropic Besov spaces.