General Relativity in asymptotically flat spacetimes gives rise to an infinite number of symmetries which form the celebrated BMS group comprising superrotations and supertranslations. These symmetries are closely related to soft theorems of gravitational scattering amplitudes. Recently it was shown that supertranslations and superrotations are only the lowest levels of a whole tower of symmetries of tree level gravitational scattering amplitudes that form a w_{1+\infty} algebra. The fate of this symmetry once loop effects are taken into account is currently unknown. In this talk I will review the emergence of this symmetry algebra based on the celestial CFT program and argue that the w_{1+\infty} algebra persists quantum corrections in self-dual gravity. This talk is based on 2111.10392 with A.Ball, S. Narayanan, and A. Strominger.