This paper reviews the recent advancements in numerical relativity (NR) simulations that have resolved gravitational memory effects and utilized BMS (Bianchi-Marklund-Strong) symmetries. Gravitational memory effects, which are nonlinear predictions of general relativity (GR), involve the net displacement experienced by freely-falling observers due to transient gravitational waves. These effects are tied to the BMS group, a larger symmetry group of future null infinity compared to the usual Poincaré group. The paper highlights how controlling the BMS freedoms in NR simulations can significantly improve the accuracy and robustness of gravitational wave (GW) models used by detectors. The introduction emphasizes the importance of testing GR through analyses of GWs from binary black hole mergers (BBHs). NR simulations are crucial for calculating these GWs, but they face challenges due to the complexity of the equations and the need for high numerical resolution. The paper discusses the limitations of perturbation theory and the importance of accurate coordinate systems, such as Bondi gauge, to study asymptotic data. The review covers the history of memory effects, the development of the BMS group, and the mathematical formalism behind BMS transformations. It explains how these symmetries lead to conservation laws, particularly supertranslation conservation laws, which are essential for understanding memory effects. The paper also discusses the impact of BMS transformations on gravitational wave shear and Weyl scalars, and how these transformations can be applied to NR waveforms. Finally, the paper reviews recent advancements in NR simulations that have resolved memory effects, including the use of Cauchy-characteristic evolution (CCE) and the BMS frame fixing program. These developments have improved the accuracy of NR waveforms and enabled more precise comparisons with post-Newtonian (PN) waveforms. The paper concludes by emphasizing the potential of memory effects and BMS symmetries to advance our understanding of GR and astrophysics.This paper reviews the recent advancements in numerical relativity (NR) simulations that have resolved gravitational memory effects and utilized BMS (Bianchi-Marklund-Strong) symmetries. Gravitational memory effects, which are nonlinear predictions of general relativity (GR), involve the net displacement experienced by freely-falling observers due to transient gravitational waves. These effects are tied to the BMS group, a larger symmetry group of future null infinity compared to the usual Poincaré group. The paper highlights how controlling the BMS freedoms in NR simulations can significantly improve the accuracy and robustness of gravitational wave (GW) models used by detectors. The introduction emphasizes the importance of testing GR through analyses of GWs from binary black hole mergers (BBHs). NR simulations are crucial for calculating these GWs, but they face challenges due to the complexity of the equations and the need for high numerical resolution. The paper discusses the limitations of perturbation theory and the importance of accurate coordinate systems, such as Bondi gauge, to study asymptotic data. The review covers the history of memory effects, the development of the BMS group, and the mathematical formalism behind BMS transformations. It explains how these symmetries lead to conservation laws, particularly supertranslation conservation laws, which are essential for understanding memory effects. The paper also discusses the impact of BMS transformations on gravitational wave shear and Weyl scalars, and how these transformations can be applied to NR waveforms. Finally, the paper reviews recent advancements in NR simulations that have resolved memory effects, including the use of Cauchy-characteristic evolution (CCE) and the BMS frame fixing program. These developments have improved the accuracy of NR waveforms and enabled more precise comparisons with post-Newtonian (PN) waveforms. The paper concludes by emphasizing the potential of memory effects and BMS symmetries to advance our understanding of GR and astrophysics.