The ongoing pandemic of severe acute respiratory syndrome (SARS), caused by the SARS-CoV-2 human coronavirus (HCoV), has brought the international scientific community before a state of emergency that needs to be addressed with intensive research for the discovery of pharmacological agents with antiviral activity. Potential antiviral natural products (NPs) have been discovered from plants of the global biodiversity, including extracts, compounds and categories of compounds with activity against several viruses of the respiratory tract such as HCoVs. However, the scarcity of natural products (NPs) and small-molecules (SMs) used as antiviral agents, especially for HCoVs, is notable. This is a review of 203 publications, which were selected using PubMed/MEDLINE, Web of Science, Scopus, and Google Scholar, evaluates the available literature since the discovery of the first human coronavirus in the 1960s; it summarizes important aspects of structure, function, and therapeutic targeting of HCoVs as well as NPs (19 total plant extracts and 204 isolated or semi-synthesized pure compounds) with anti-HCoV activity targeting viral and non-viral proteins, while focusing on the advances on the discovery of NPs with anti-SARS-CoV-2 activity, and providing a critical perspective.