Coronaviruses have a broad host range and exhibit high zoonotic potential. In this chapter, we describe their genomic organization in terms of encoded proteins and provide an introduction to the peculiar discontinuous transcription mechanism. Further, we present evolutionary conserved genomic RNA secondary structure features, which are involved in the complex replication mechanism. With a focus on computational methods, we review the emergence of SARS-CoV-2 starting with the 2019 strains. In that context, we also discuss the debated hypothesis of whether SARS-CoV-2 was created in a laboratory. We focus on the molecular evolution and the epidemiological dynamics of this recently emerged pathogen and we explain how variants of concern are detected and characterised. COVID-19, the disease caused by SARS-CoV-2, can spread through different transmission routes and also depends on a number of risk factors. We describe how current computational models of viral epidemiology, or more specifically, phylodynamics, have facilitated and will continue to enable a better understanding of the epidemic dynamics of SARS-CoV-2.