We consider a relevant generalization of the standard Gevrey classes, the so-called multi-anisotropic spaces, defined in terms of a given complete polyhedron. With respect to the previous literature on the subject, we concentrate here in the study of the topology. It is defined as inductive and projective limit of Banach spaces, in two equivalent ways, based on the estimates on the derivatives and on the Fourier transform, respectively. We consequently introduce the dual space, the class of the multi-anisotropic ultradistributions, of which we give different characterizations, study topological and algebraic properties and present some applications.