The search for pragmatic observables of quantum gravity remains at the forefront of fundamental physics research. A large set of ideas collectively known as the gauge-gravity duality have proven fruitful in tackling this problem. While such a duality is believed to universally govern gravitational theories, its nature in theories of gravity that describe our universe to a good degree of approximation is still little understood. In the first part of this talk I will describe recent progress in formulating a holographic correspondence for gravity in four-dimensional asymptotically flat spacetimes. Celestial amplitudes obtained by recasting the gravitational S-matrix in a basis of asymptotic boost eigenstates are candidate dual observables living in two dimensions. I will describe their properties highlighting the features that allow for powerful conformal field theory (CFT) methods to be used in the study of gravity. As an application of this framework, I will show how non-perturbative features of quantum gravity are encoded in the analytic structure of 4-point celestial amplitudes. In the second part I will discuss the rich symmetry structure of celestial CFT and argue that it can be used to infer non-trivial information about four-dimensional gravitational dynamics. In particular, I will demonstrate that a recently discovered tower of celestial soft symmetries implies a set of dynamical evolution equations for an infinity of asymptotic charges in general relativity. I will finally provide evidence that these charges realize a representation of a higher-spin symmetry algebra on the gravitational phase space.