  In the first part of this thesis, we study the Yamabe problem with
singularities, that we can announce as follow: Given a compact Riemannian
manifold $(M,g)$, find a constant scalar curvature metric, conformal to $g$,
when $g$ has not necessarily the usual regularity (it can be $C^1$). To solve
this problem, we start the study of the Yamabe type equations. We show that all
the known properties in the smooth case are still valid. Under some
assumptions, we prove the existence and uniqueness of solutions. The second
part is dedicated to the Hebey-Vaugon conjecture, stated in their paper about
the equivariant Yamabe problem. We prove that this conjecture is true in some
new cases, after we generalize T. Aubin's theorem.
