We review approaches aimed at deciphering the physical mechanisms responsible for viral structure and assembly. We discuss the basic principles of condensed matter physics, macroscopic electrostatics and elastomechanics as they apply to nanosized two-dimensional biomolecular shells with spherical topology and icosahedral symmetry, as well as their proper extension to nonspherical structures pertinent to retroviruses. We examine the relation between the virus structure, the physical interactions that are driving their (self)assembly and the thermodynamics of transition from an isotropic protein solution to the assembled shell state, and discuss the driving forces for large-scale structural reorganizations characterizing maturation processes in the already assembled nano-shells. We furthermore review physical models corresponding to the condensed states of confined genome-carrying biopolymers in viral nano-shells during virus self-assembly and host-cell infection processes, and show how the combination of discrete and continuum coarse grained mechanics, commonly used in the fundamental physical description of viruses, together with the pertinent description of generic long-range electrostatic and specific short-range interactions give insight into the details of structural order and mechanical properties of viruses and elucidates their role in nano-container and nano-machine functions.