Conformal maps are homeomorphisms which preserve the shapes of infinitesimally small objects. By the classical Uniformization theorem, every smooth surface that admits a homeomorphism onto the standard two-sphere actually admits a conformal homeomorphism onto the standard two-sphere. The Non-smooth uniformization problem asks for strongest possible extensions when smooth surfaces are replaced with metric spaces. We present recent developments on metric surfaces that have finite (Hausdorff) area, and discuss connections to classical problems on conformal maps.