Symmetries provide valuable insights into theories of physics. In recent years, analyses of the symmetries of general relativity -- our best tested theory of gravity -- have led to remarkable advances in our understanding of the nature of gravity in our universe. It is the goal of this thesis to describe some of these advances. A technical summary of the contents of this work is as follows. In this thesis, the algebra of symmetries and the charges associated with null and spatial boundaries in four-dimensional spacetimes in general relativity are rigorously derived. All reported work herein is in the context of asymptotic boundaries in asymptotically flat spacetimes. In addition, previously conjectured "matching" relations between the symmetries and charges associated with past and future null infinity in asymptotically flat spacetimes are proven. Moreover, proposals for extensions of the algebra of asymptotic symmetries are also studied. Finally, to analyze the implications of asymptotic symmetries and the associated charges for classical scattering processes, a detailed study of the low energy dynamics of a classical (complex) scalar field coupled to electromagnetism on Minkowski spacetime is also included.