The quasiconformal Jacobian problem asks for a
characterization of those nonnegative locally integrable functions that
are comparable to the Jacobian determinant of a quasiconformal map. It
turns out that in dimension n=2 this problem is equivalent to the problem
of characterizing metric spaces that are bi-Lipschitz equivalent to the
Euclidean plane. I will explain this connection, present some partial
results, and pose open problems.