  In this article, we prove several results about the extension to the boundary
of conformal immersions from an open subset $\Omega$ of a Riemannian manifold
$L$, into another Riemannian manifold $N$ of the same dimension. In dimension
$n \geq 3$, and when the $(n-1)$-dimensional Hausdorff measure of $\partial
\Omega$ is zero, we completely classify the cases when $\partial \Omega$
contains essential singular points, showing that $L$ and $N$ are conformally
flat and making the link with the theory of Kleinian groups.
