Carleson measures and vanishing Carleson measures have been well researched in harmonic analysis, and appear frequently recently in the study of the theory of BMO-Teichmuller spaces from the viewpoint of complex analysis . We address some recent results concerning Carleson measures and their invariance under pull-back and push-forward operators. Moreover, the analogous statements for vanishing Carleson measures will also be presented. As an application, we give that some quotient space of the space of chord-arc curves has a natural complex Banach manifold structure. This talk is mainly based on a joint work with Professor Michel Zinsmeister.

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