Coronaviruses are single-stranded RNA viruses that affect humans and a wide variety of animal species, including livestock, wild animals, birds, and pets. These viruses have an affinity for different tissues, such as those of the respiratory and gastrointestinal tract of most mammals and birds and the hepatic and nervous tissues of rodents and porcine. As coronaviruses target different host cell receptors and show divergence in the sequences and motifs of their structural and accessory proteins, they are classified into groups, which may explain the evolutionary relationship between them. The interspecies transmission, zoonotic potential, and ability to mutate at a higher rate and emerge into variants of concern highlight their importance in the medical and veterinary fields. The contribution of various factors that result in their evolution will provide better insight and may help to understand the complexity of coronaviruses in the face of pandemics. In this review, important aspects of coronaviruses infecting livestock, birds, and pets, in particular, their structure and genome organization having a bearing on evolutionary and zoonotic outcomes, have been discussed.