  We provide an account of the construction of the moduli stack of elliptic
curves as an analytic orbifold. While intimately linked to Thurston's point of
view on the subject (discrete groups acting properly and effectively on
differentiable manifolds), the construction of the modular orbi-curve and its
universal family of elliptic curves ends up requiring a bit more technology, in
order to allow for non-effective actions. The paper is entirely expository and
makes no claims to originality: its main goal is to be self-contained enough in
order to be useful to young researchers who are entering the field and are
interested in the interactions between differential and algebraic geometry.
