The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the etiological agent responsible for the coronavirus disease 2019 (COVID-19). The high rate of mutation of this virus is associated with a quick emergence of new viral variants that have been rapidly spreading worldwide. Several mutations have been documented in the receptor-binding domain (RBD) of the viral spike protein that increases the interaction between SARS-CoV-2 and its cellular receptor, the angiotensin-converting enzyme 2 (ACE2). Mutations in the spike can increase the viral spread rate, disease severity, and the ability of the virus to evade either the immune protective responses, monoclonal antibody treatments, or the efficacy of current licensed vaccines. This review aimed to highlight the functional virus classification used by the World Health Organization (WHO), Phylogenetic Assignment of Named Global Outbreak (PANGO), Global Initiative on Sharing All Influenza Data (GISAID), and Nextstrain, an open-source project to harness the scientific and public health potential of pathogen genome data, the chronological emergence of viral variants of concern (VOCs) and variants of interest (VOIs), the major findings related to the rate of spread, and the mutations in the spike protein that are involved in the evasion of the host immune responses elicited by prior SARS-CoV-2 infections and by the protection induced by vaccination.