Since the emergence of the coronavirus disease 2019 (COVID-19) pandemic, several different variants and strains of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) have been identified. Each of these variants has several mutations in different constituents of the original virus, such as the spike (S) glycoprotein, receptor-binding domain, N-terminal domain, and furin cleavage site region. These mutations mainly influence the virulence, infectivity, and transmission of the virus. Also, they can help the virus escape the natural- or vaccine-induced immunity in the host body. The Centers for Disease Control and Prevention categorized these variants into three major classes: variants of interest, variants of concern (VOC), and variants of high consequence. In this review, four VOCs, i.e., B.1.1.7, B.1.351, P.1, and B.1.427/B.1.429, have been presented, and their potential complications, pathogenicity, transmissibility, and capability of escaping natural- or vaccine-induced immunity are discussed. Moreover, the novel B.1.617 variant and its known characteristics are also demonstrated. In conclusion, this review can help clinicians and scientists better understand the most critical properties of the mentioned concerning variants of SARS-CoV-2 and guide them to conduct future studies on new preventive and therapeutic approaches in fighting COVID-19.