  We study the Carleson measures associated to the Hardy and weighted Bergman
spaces defined on general simply connected domains. This program was initiated
by Zinsmeister in his paper \textit{Les domaines de Carleson }(1989), where he
shows that the geometry of the domain plays a fundamental role. We will review
the classical results presenting, in some cases, alternative proofs and will
examine the situation for the weighted Bergman spaces.
