The classical uniformization theorem states that every simply connected Riemann surface is conformally equivalent to the unit disc, the complex plane, or the Riemann sphere. In this talk we are interested in non-smooth generalizations of this statement, where conformality is replaced by quasisymmetry and (weak) quasiconformality. Our goal is to demonstrate that no additional assumptions beyond local finiteness of area are needed to establish the existence of a suitable parametrization of a general metric surface.