The COVID-19 pandemic caused by SARS-CoV-2 infection is spreading at an alarming rate and has created an unprecedented health emergency around the globe. There is no effective vaccine or approved drug treatment against COVID-19 and other pathogenic coronaviruses. The development of antiviral agents is an urgent priority. Biochemical events critical to the coronavirus replication cycle provided a number of attractive targets for drug development. These include, spike protein for binding to host cell-surface receptors, proteolytic enzymes that are essential for processing polyproteins into mature viruses, and RNA-dependent RNA polymerase for RNA replication. There has been a lot of ground work for drug discovery and development against these targets. Also, high-throughput screening efforts have led to the identification of diverse lead structures, including natural product-derived molecules. This review highlights past and present drug discovery and medicinal-chemistry approaches against SARS-CoV, MERS-CoV and COVID-19 targets. The review hopes to stimulate further research and will be a useful guide to the development of effective therapies against COVID-19 and other pathogenic coronaviruses.