Gravitational memory effects and BMS symmetries at future null infinity have been integrated into numerical relativity (NR) simulations, significantly improving gravitational wave (GW) models and our understanding of general relativity (GR). This review outlines the history and intuition behind memory effects and BMS symmetries, their manifestations in GWs, and how controlling BMS freedoms enhances waveform models used by GW detectors. Memory effects, nonlinear predictions of GR, represent net displacement of freely-falling observers due to transient GWs. They are tied to the BMS group, a larger symmetry than the Poincaré group, and are crucial for testing GR and understanding the asymptotic structure of spacetime. The BMS group includes Lorentz transformations and supertranslations, which are angle-dependent spacetime translations. Supertranslations are essential for understanding memory effects and are linked to conservation laws. The Bondi gauge provides a coordinate system adapted to inertial observers, reducing coordinate ambiguities in GW waveforms. By fixing BMS frame, numerical relativity simulations can produce more accurate and robust GW models. The review discusses the mathematical foundations of BMS symmetries, their connection to memory effects, and the role of supertranslations in conservation laws. It also highlights the importance of BMS frame fixing in ensuring accurate waveform modeling. The paper emphasizes the significance of memory effects in testing GR and their potential to reveal new astrophysical information. Future research in numerical relativity, memory effects, and gravity testing is expected to benefit from these advancements.Gravitational memory effects and BMS symmetries at future null infinity have been integrated into numerical relativity (NR) simulations, significantly improving gravitational wave (GW) models and our understanding of general relativity (GR). This review outlines the history and intuition behind memory effects and BMS symmetries, their manifestations in GWs, and how controlling BMS freedoms enhances waveform models used by GW detectors. Memory effects, nonlinear predictions of GR, represent net displacement of freely-falling observers due to transient GWs. They are tied to the BMS group, a larger symmetry than the Poincaré group, and are crucial for testing GR and understanding the asymptotic structure of spacetime. The BMS group includes Lorentz transformations and supertranslations, which are angle-dependent spacetime translations. Supertranslations are essential for understanding memory effects and are linked to conservation laws. The Bondi gauge provides a coordinate system adapted to inertial observers, reducing coordinate ambiguities in GW waveforms. By fixing BMS frame, numerical relativity simulations can produce more accurate and robust GW models. The review discusses the mathematical foundations of BMS symmetries, their connection to memory effects, and the role of supertranslations in conservation laws. It also highlights the importance of BMS frame fixing in ensuring accurate waveform modeling. The paper emphasizes the significance of memory effects in testing GR and their potential to reveal new astrophysical information. Future research in numerical relativity, memory effects, and gravity testing is expected to benefit from these advancements.