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Dynamical and structural systems are susceptible to sudden excitations and loadings such as wind gusts, blasts, earthquakes, and others which may cause destructive vibration amplitudes and lead to catastrophic impact on human lives and economy. Therefore, various vibration absorbers of linear and nonlinear coupling dynamics have been widely studied in plenty of publications where some have been applied in real-world practical applications. Firstly, the tuned-mass-damper (TMD), the first well-known linear vibration absorber that has been well-studied in the literature and applied with various structural and dynamical systems, is discussed. The linear vibration absorbers such as TMDs are widely used in real-life small- and large-scale structures due to their robust performance in vibration suppression of the low natural frequency structural modes. However, the TMD performs efficiently at narrowband frequency range where its performance is deteriorated by any changes in the frequency content in the structure and the TMD itself. Therefore, the targeted-energy-transfer mechanism which is found to be achieved by nonlinear energy sinks (NESs) has ignited the interest in passive nonlinear vibration suppression. Unlike TMDs, the NESs are dynamical vibration absorbers that achieve vibration suppression for wide range of frequency-energy levels. Given the very rapid growth in this field and the extensive research studies supporting the robustness of the NESs, this paper presents the different types of NESs and their applications with main emphasis on the rotary-based and impact-based NESs since they are of high impact in the literature due to their strong nonlinear dynamical behavior and robust targeted energy transfer.