Explore a mathematical lecture examining non-smooth generalizations of the classical uniformization theorem for Riemann surfaces, delivered at the Erwin Schrödinger International Institute's Thematic Programme on "Infinite-dimensional Geometry." Delve into how conformality can be replaced by quasisymmetry and weak quasiconformality when studying metric surface parametrization. Learn why local finiteness of area is sufficient for establishing suitable parametrizations of general metric surfaces, without requiring additional assumptions. The 40-minute presentation bridges classical Riemann surface theory, which states that every simply connected Riemann surface is conformally equivalent to the unit disc, complex plane, or Riemann sphere, with modern approaches to metric surface uniformization.