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.