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The Bondi-Metzner-Sachs-van der Burg (BMS) group is the asymptotic symmetry group of radiating asymptotically flat spacetimes. It has recently received renewed interest in the context of the flat holography and the infrared structure of gravity. In this thesis, we investigate the consequences of considering extensions of the BMS group in four dimensions with superrotations. In particular, we apply the covariant phase space methods on a class of first order gauge theories that includes the Cartan formulation of general relativity and specify this analysis to gravity in asymptotically flat spacetime. Furthermore, we renormalize the symplectic structure at null infinity to obtain the generalized BMS charge algebra associated with smooth superrotations. We then study the vacuum structure of the gravitational field, which allows us to relate the so-called superboost transformations to the velocity kick/refraction memory effect. Afterward, we propose a new set of boundary conditions in asymptotically locally (A)dS spacetime that leads to a version of the BMS group in the presence of a non-vanishing cosmological constant, called the $Λ$-BMS asymptotic symmetry group. Using the holographic renormalization procedure and a diffeomorphism between Bondi and Fefferman-Graham gauges, we construct the phase space of $Λ$-BMS and show that it reduces to the one of the generalized BMS group in the flat limit.