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In this talk, I will review boundary symmetries of asymptotically flat spacetimes, their relation to soft theorems and their realization in the recently-proposed celestial CFTs. It has long been known that the boundary symmetries of asymptotically flat spacetimes are much richer than just the Poincaré algebra. Over the last decade, the corresponding BMS algebra and several of its extensions have been related to identities satisfied by amplitudes involving `soft’ particles. These insights led to a proposal for a holographic description of physics in 4d asymptotically flat spacetimes in terms of a peculiar type of 2d CFT on the celestial sphere. I will review some of the motivation for this proposal, and I will discuss recent work on an infinite tower of (sub)^n-leading symmetries and their mapping to a w1+\infty algebra.