Let \(S\) be the set of all mappings \(\text{Log} f'(z)\), where \(f\) is conformal in the unit disk \(\mathbb{D}\) of the complex plane \(\mathbb{C}\). Then \(S\) is a bounded subset of the Bloch space \({\mathcal B}\), the set of \(\varphi\) holomorphic in \(\mathbb{D}\) with \(\| \varphi \|_{\mathcal B}= \sup\{(1- | z|^2) |\varphi'(z) |;\;z\in\mathbb{D}\} <\infty\). Next, \(T(1)\), the interior of \(S\) in \({\mathcal B}\), is a model of the universal Teichmüller space which can be characterized by the following set of equivalent conditions: (1) \(\text{Log} f' \in T(1)\); (2) \(f\) has a quasiconformal extension to \(\mathbb{C}\); (3) \(\Omega =f(\mathbb{D})\) is a \(C\)-quasidisk, i.e., a Jordan domain such that \(\text{diam} (\gamma)\leq C| z-\zeta |\) for \(z,\zeta\in \partial \Omega\), where \(\gamma\) is the smaller subarc of \(\partial \Omega \smallsetminus \{z,\zeta\}\); (4) \(h=f^{-1} \circ g\) is quasisymmetric for each \(g\) which maps \(C\smallsetminus \overline \mathbb{D}\) conformally onto \(\mathbb{C} \smallsetminus \overline {f(\mathbb{D})}\). In this paper, the authors introduce a different ``Teichmüller theory'' in the BMO topology, with \(S\) replaced by \(\Sigma= S\cap BMOA (\mathbb{D})\), for which the following elegant set of equivalent conditions hold: (a) \(\text{Log} f'\in BMOA (\mathbb{D}) \cap T(1)\); (b) \(f\) has a quasiconformal extension to \(\mathbb{C}\) with dilatation \(\mu\) such that \(|\mu |^2 (| z|^2-1)^{-1} dx dy\) is a Carleson measure in \(\mathbb{C} \smallsetminus \overline \mathbb{D}\); (c) \(\Omega= f(\mathbb{D})\) is a quasidisk each point \(z\) of which lies in a \(C\)-Lavrent'ev domain \(\Omega_z \subset \Omega\) with \(\text{diam} (\Omega_z)\) and \(\Lambda (\partial \Omega \cap \partial \Omega_z)\) comparable to \(d(z,\partial \Omega)\); (d) \(h=f^{-1} \circ g\) is strongly quasisymmetric for each \(g\) which maps \(\mathbb{C} \smallsetminus \overline \mathbb{D}\) conformally onto \(\mathbb{C} \smallsetminus \overline {f(\mathbb{D})}\). The geometric condition in (3) is due to Bishop and Jones.