In this paper, the crucial results on quasiconformal mappings are two distortion theorems. The first gives a Hölder estimate and the second a control over the measure distortion for mappings with small BMO-norm \(\| \log J_ f\|_*\) \((J_ f\) is the Jacobian determinant of f). The authors then pove the following theorem: For all \(K<2\) there exists a positive constant \(b=b(K)\) such that any locally K-quasiconformal mapping f in a disk (or a halfplane) with \(\| \log J_ f\|_*\leq b\) is injective. Examples of locally 2-quasiconformal mappings with arbitrarily small BMO-norm \(\| \log J_ f\|_*\) which fail to be injective are exhibited. Note that in higher dimensions a similar theorem holds for locally K-quasiconformal mappings with small K. In this case the hypothesis on \(\| \log J_ f\|_*\) is redundant [\textit{J. Sarvas}, Duke Math. J. 43, 147-158 (1976; Zbl 0357.30016)]. For K-quasiconformal plane mappings with small K, the control over the size of \(\| \log J_ f\|_*\) appears to be a sufficiently strong tool for the characterization of the quasidisks (theorems 1.3B and 1.4B). Analytic mappings f in the unit disk D with f'\(\neq 0\) satisfy \[ \| \log J_ f\|_*\leq \sup_{z\in D}| f''/f'| (1-| z|^ 2)\leq 6\| \log J_ f\|_*. \] This leads to a comparison for theorems on analytic functions f involving the boundedness condition for \(| f''/f'| (1-| z|^ 2)\) and theorems on quasiconformal mappings where the size of \(\| \log J_ f\|_*\) intervenes. The universal Teichmüller space T is defined as the subset of \[ S=\{S_ f=(f''/f')'-(f''/f')^ 2: f\quad conformal\quad in\quad D\} \] consisting of those elements \(S_ f\) for which f has a quasiconformal extension to \({\mathbb{C}}\). A major part of the paper is devoted to the investigation of the analogously defined spaces \(T_ 1\) and \(S_ 1\) \[ S_ 1=\{f''/f': f\quad conformal\quad in\quad D\}. \] These spaces are embedded in the Banach space \[ E_ 1=\{\phi \quad analytic\quad in\quad D: \| \phi \| =\sup_{z\in D}| f''/f'| (1-| z|^ 2)<\infty \}. \] It is shown that \(T_ 1=int S_ 1\), \(S_ 1\setminus T_ 1\neq \emptyset\). The results are used for a refined analysis of the boundary of the Teichmüller space T.