  We study quasiconformal mappings in planar domains $\Omega$ and their
regularity properties described in terms of Sobolev, Bessel potential or
Triebel-Lizorkin scales. This leads to optimal conditions, in terms of the
geometry of the boundary $\partial \Omega$ and of the smoothness of the
Beltrami coefficient, that guarantee the global regularity of the mappings in
these classes. In the Triebel-Lizorkin class with smoothness below $1$, the
same conditions give global regularity in $\Omega$ for the principal solutions
with Beltrami coefficient supported in $\Omega$.
