The aim of this paper is to put a nice analytic structure on the higher-dimensional Teichm\"uller and Riemann spaces, which are only defined as topological spaces. We show that they can be described as Artin analytic stacks of families of complex manifolds, that is admitting a presentation as a groupoid whose objects and morphisms form a finite-dimensional analytic space and whose source and target maps are smooth morphisms. This is achieved under the sole condition that the dimension of the automorphism group of each structure is bounded by a fixed integer. The main point here is the existence of a global (multi)-foliated structure on the space of complex operators whose holonomy data encodes how to glue the local Kuranishi spaces to obtain a groupoid presentation of the Teichm\"uller space.