Molecular dynamics (MD) simulations are powerful theoretical methods that can reveal biomolecular properties, such as structure, fluctuations, and ligand binding, at the level of atomic detail. In this review article, recent MD simulation studies on these biomolecular properties of the RNA-dependent RNA polymerase (RdRp), which is a multidomain protein, of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) are presented. Although the tertiary structures of RdRps in SARS-CoV-2 and SARS-CoV are almost identical, the RNA synthesis activity of RdRp of SARS-CoV is higher than SARS-CoV-2. Recent MD simulations observed a difference in the dynamic properties of the two RdRps, which may cause activity differences. RdRp is also a drug target for Coronavirus disease 2019 (COVID-19). Nucleotide analogs, such as remdesivir and favipiravir, are considered to be taken up by RdRp and inhibit RNA replication. Recent MD simulations revealed the recognition mechanism of RdRp for these drug molecules and adenosine triphosphate (ATP). The ligand-recognition ability of RdRp decreases in the order of remdesivir, favipiravir, and ATP. As a typical recognition process, it was found that several lysine residues of RdRp transfer these ligand molecules to the binding site such as a “bucket brigade.” This finding will contribute to understanding the mechanism of the efficient ligand recognition by RdRp. In addition, various simulation studies on the complexes of SARS-CoV-2 RdRp with several nucleotide analogs are reviewed, and the molecular mechanisms by which these compounds inhibit the function of RdRp are discussed. The simulation studies presented in this review will provide useful insights into how nucleotide analogs are recognized by RdRp and inhibit the RNA replication.