Coronaviruses are extremely susceptible to genetic changes due to the characteristic features of the genome structure, life cycle and environmental pressure. Their remarkable variability means that they can infect many different species of animals and cause different disease symptoms. Moreover, in some situations, coronaviruses might be transmitted across species. Although they are commonly found in farm, companion and wild animals, causing clinical and sometimes serious signs resulting in significant economic losses, not all of them have been classified by the World Organization for Animal Health (OIE) as hazardous and included on the list of notifiable diseases. Currently, only three diseases caused by coronaviruses are on the OIE list of notifiable terrestrial and aquatic animal diseases. However, none of these three entails any administrative measures. The emergence of the SARS-CoV-2 infections that have caused the COVID-19 pandemic in humans has proved that the occurrence and variability of coronaviruses is highly underestimated in the animal reservoir and reminded us of the critical importance of the One Health approach. Therefore, domestic and wild animals should be intensively monitored, both to broaden our knowledge of the viruses circulating among them and to understand the mechanisms of the emergence of viruses of relevance to animal and human health.