Given a metric space homeomorphic to a Euclidean space (or a
sphere, for example) it is natural to ask for geometric parametrization
with a homeomorphisms which respect the metric, for example, bilipschitz
or quasisymmetric maps.

In 1996 Semmes constructed metric spaces which are homeomorphic to the
standard Euclidean 3-space but not admit geometric parametrizations.
The construction was surprising, since the metric space is close to a
Euclidean space from the point of view of measure and geometry. In 2004
Bonk and Kleiner showed that metric two dimensional spheres satisfying
the same conditions are in fact quasisymmetric to the standard sphere.

Although there are no good geometric homeomorphisms from Semmes' space
to the Euclidean space, there exist other natural geometric mappings to
the Euclidean space by results of Heinonen and Rickman in 2004. These
mappings give Semmes' space a natural ``branched parametrization''.

Semmes' construction is based on a classical geometric topology. In this
talk I will discuss the joint work with Jang-Mei Wu on general framework
of geometric decomposition spaces and their quasisymmetric
(non-)parametrizability. I will also discuss recent work with Kai Rajala
and Jang-Mei on the existence of branched parametrizations in this
general setting.