Viral quasispecies refers to the complex and dynamic mutant distributions (also termed mutant spectra, clouds, or swarms) that arise as a result of high error rates during RNA genome replication. The mutant spectrum of individual RNA virus populations is modified by continuous generation of variant genomes, competition and interactions among them, environmental influences, bottleneck events, and bloc transmission of viral particles. Quasispecies dynamics provides a new perspective on how viruses adapt, evolve, and cause disease, and sheds light on strategies to combat them. Molecular flexibility, together with ample opportunity of mutant cloud traffic in our global world, are key ingredients of viral disease emergences, as exemplified by the recent COVID-19 pandemic. In the present article, we present a brief overview of the molecular basis of mutant swarm formation and dynamics, and how the latter relates to viral disease and epidemic spread. We outline future challenges derived of the highly diverse cellular world in which viruses are necessarily installed.