  This paper is a comprehensive introduction to the results of [7]. It grew as
an expanded version of a talk given at INdAM Meeting Complex and Symplectic
Geometry, held at Cortona in June 12-18, 2016. It deals with the construction
of the Teichm\"uller space of a smooth compact manifold M (that is the space of
isomorphism classes of complex structures on M) in arbitrary dimension. The
main problem is that, whenever we leave the world of surfaces, the
Teichm\"uller space is no more a complex manifold or an analytic space but an
analytic Artin stack. We explain how to construct explicitly an atlas for this
stack using ideas coming from foliation theory. Throughout the article, we use
the case of $\mathbb{S}^3\times\mathbb{S}^1$ as a recurrent example.
