  We establish uniformization results for metric spaces that are homeomorphic
to the euclidean plane or sphere and have locally finite Hausdorff 2-measure.
Applying the geometric definition of quasiconformality, we give a necessary and
sufficient condition for such spaces to be QC equivalent to the euclidean
plane, disk, or sphere. Moreover, we show that if such a QC parametrization
exists, then the dilatation can be bounded by 2. As an application, we show
that the euclidean upper bound for measures of balls is a sufficient condition
for the existence of a 2-QC parametrization. This result gives a new approach
to the Bonk-Kleiner theorem on parametrizations of Ahlfors 2-regular spheres by
quasisymmetric maps.
