Despite the wealth of data describing the ecological factors that underpin viral emergence, little is known about the evolutionary processes that allow viruses to jump species barriers and establish productive infections in new hosts. Understanding the evolutionary basis to virus emergence is therefore a key research goal and many of the debates in this area can be considered within the rigorous theoretical framework established by evolutionary genetics. In particular, the respective roles played by natural selection and genetic drift in shaping genetic diversity are also of fundamental importance for understanding the nature of viral emergence. Herein, we discuss whether there are evolutionary rules to viral emergence, and especially whether certain types of virus, or those that infect a particular type of host species, are more likely to emerge than others. We stress the complex interplay between rates of viral evolution and the ability to recognize cell receptors from phylogenetically divergent host species. We also emphasize the current lack of convincing data as to whether viral emergence requires adaptation to the new host species during the early stages of infection, or whether it is largely a chance process involving the transmission of a viral strain with the necessary genetic characteristics.