  The aim of this paper is to study the structure of the higher-dimensional
Teichm\"uller and Riemann moduli spaces, viewed as stacks over the category of
complex manifolds. We first show that the space of complex operators on a
smooth manifold admits a foliation transversely modeled on a translation
groupoid, a concept that we define here. We then show how to construct
explicitly a holonomy groupoid for such a structure and show that in this case
its objects and morphisms form a finite-dimensional analytic space and its
source and target maps are smooth morphisms. This holonomy data encodes how to
glue the local Kuranishi spaces to obtain a groupoid presentation of the
Teichm\"uller and Riemann moduli stacks, which can thus be characterized as
Artin analytic stacks. This is achieved under the sole condition that the
dimension of the automorphism group of each structure is bounded by a fixed
integer.
