The paper explores some consequences of the a.e. approximability of measurable functions ofz,[zbar], by polynomials inzto the approximability of arbitrary quasiconformal mappings of a region G by Teichmüller mappings generated by analytic quadratic differentials ϕ(z). In particular, iffis a quasiconformal mapping of the unit diskD, thenfcan be approximated uniformly in Đ by such Teichmüller maps; moreover, there exists a so-calledgoodapproximation offby generalized Teichmüller maps, uniformly in Đ. On the other hand, it turns out to be possible for a sequence of Teichmüller mapsfn, to give even a good approximation of a Teichmüller mapfwithout having the corresponding quadratic differentials ϕnconverging uniformly in any neighborhood. Some of the preceding facts have their counterparts for mappings close to the identity, since the supremum ofover analytic ϕ does not differ from the supremum over measurable ϕ.