The authors study classes \({\mathcal F}^{\Phi}\) of quasiconformal mappings f of plane domains defined by conditions of the type \[ \iint_{E}\Phi (p(z),z) dx dy\leq M, \] where \(p(z)=(1+| \mu (z)|)/(1-| \mu (z)|)\) and \(\mu\) is the complex dilatation of f. Assuming some regularity of \(\Phi\) they state convergence and compactness results and give a variational formula by which properties of extremals for certain functionals \(\Omega\) in \({\mathcal F}^{\Phi}\) can be described.