Icosahedral viruses are among the smallest and simplest of biological systems. The investigation of their structures represented the first step toward the establishment of molecular biophysics, over half a century ago. Many research groups are now pursuing investigations of viral assembly, a process that could offer new opportunities for the design of antiviral drugs and novel nanoparticles. A variety of experimental, theoretical and computational methods have been brought to bear on the study of virus structure and assembly. In this Perspective we review the contributions of theoretical and computational approaches to our understanding of the structure, energetics, thermodynamics and assembly of DNA bacteriophage and single-stranded icosahedral RNA viruses.