With few exceptions, the shells (capsids) of sphere-like viruses have the symmetry of an icosahedron and are composed of coat proteins (subunits) assembled in special motifs, the T -number structures. Although the synthesis of artificial protein cages is a rapidly developing area of materials science, the design criteria for self-assembled shells that can reproduce the remarkable properties of viral capsids are only beginning to be understood. We present here a minimal model for equilibrium capsid structure, introducing an explicit interaction between protein multimers (capsomers). Using Monte Carlo simulation we show that the model reproduces the main structures of viruses in vivo ( T -number icosahedra) and important nonicosahedral structures (with octahedral and cubic symmetry) observed in vitro . Our model can also predict capsid strength and shed light on genome release mechanisms.