Let \(f\) denote an ACL sense-preserving open and discrete mapping defined in a domain of the complex plane (where ACL stands for absolutely continuous on lines). Then the complex dilatation \(\mu(z)=\overline \partial f(z)/ \partial f(z)\) with \(|\mu |<1\) is defined a.e. and the dilatation \(K(z)= (1+|\mu(z)|)/(1- \mu(z)|)\) is finite a.e. The mapping \(f\) is called BMO-quasiregular if \(K(z)\) is bounded by some BMO function (which means a function of bounded mean oscillation), it is called BMO-quasiconformal if it is, in addition, homeomorphic. The basic properties of these mappings are now studied starting with results on integrability, distortion and convergence. Then existence, uniqueness and representation theorems on the Beltrami equation, under the dilatation condition described above, are proved and the results are compared with related ones obtained by David, Tukia, and more recently by Brakalova and Jenkins and also by Astala, Iwaniec, Koskela and Martin. The interesting paper closes with theorems on removability, reflection principle, boundary behavior and mapping properties.