The coronavirus disease 2019 (COVID-19) pandemic is caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), a recently emerged human coronavirus. COVID-19 vaccines have proven to be successful in protecting the vaccinated from infection, reducing the severity of disease, and deterring the transmission of infection. However, COVID-19 vaccination faces many challenges, such as the decline in vaccine-induced immunity over time, and the decrease in potency against some SARS-CoV-2 variants including the recently emerged Omicron variant, resulting in breakthrough infections. The challenges that COVID-19 vaccination is facing highlight the importance of the discovery of antivirals to serve as another means to tackle the pandemic. To date, neutralizing antibodies that block viral entry by targeting the viral spike protein make up the largest class of antivirals that has received US FDA emergency use authorization (EUA) for COVID-19 treatment. In addition to the spike protein, other key targets for the discovery of direct-acting antivirals include viral enzymes that are essential for SARS-CoV-2 replication, such as RNA-dependent RNA polymerase and proteases, as judged by US FDA approval for remdesivir, and EUA for Paxlovid (nirmatrelvir + ritonavir) for treating COVID-19 infections. This review presents an overview of the current status and future direction of antiviral drug discovery for treating SARS-CoV-2 infections, covering important antiviral targets such as the viral spike protein, non-structural protein (nsp) 3 papain-like protease, nsp5 main protease, and the nsp12/nsp7/nsp8 RNA-dependent RNA polymerase complex.