While capsid proteins are assembled around single-stranded genomic DNA or RNA in rod-shaped viruses, the lengthy double-stranded genome of other viruses is packaged forcefully within a preformed protein shell. This entropically unfavourable DNA or RNA packaging is accomplished by an ATP-driven viral nanomotor, which is mainly composed of two components, the oligomerized channel and the packaging enzymes. This intriguing DNA or RNA packaging process has provoked interest among virologists, bacteriologists, biochemists, biophysicists, chemists, structural biologists and computational scientists alike, especially those interested in nanotechnology, nanomedicine, AAA+ family proteins, energy conversion, cell membrane transport, DNA or RNA replication and antiviral therapy. This review mainly focuses on the motors of double-stranded DNA viruses, but double-stranded RNA viral motors are also discussed due to interesting similarities. The novel and ingenious configuration of these nanomotors has inspired the development of biomimetics for nanodevices. Advances in structural and functional studies have increased our understanding of the molecular basis of biological movement to the point where we can begin thinking about possible applications of the viral DNA packaging motor in nanotechnology and medical applications.