Viruses are submicroscopic biological entities that need to infect a host cell in order to replicate. In their simplest form viruses are constituted by an infective genetic material and a protein shell (the capsid) that protects the viral genome. In this thesis we try to elucidate the general physical principles playing a major role in the morphology, stability, and assembly of viral capsids. Therefore, in the first part of the thesis, we develop a general theory that characterizes spherical and bacilliform (or prolate) capsids based on icosahedral symmetry under the same geometrical framework. In addition, we demonstrate that the structures derived in this geometrical study are obtained spontaneously from the free energy minimization of a very generic interaction among the viral capsomers of the capsid. In the second part of the thesis, we analyze the role of the discrete nature of capsids and the organization of capsomers in the actual mechanical properties of shells. We show that the icosahedral class P influences the stability and mechanical response of the quasi-spherical capsids. We also determine that under expansion, spherical shells tend to produce polyhedral structures (buckling), which are more resistant, and it is in consonance with the maturation process observed in some viruses. We also unveil the existence of built-in stress in the empty procapsids of the elongated bacteriophage φ29. This phenomenon is intimately related to the discrete nature of the structure, and reinforces the mechanical properties of the shell inverting the classical anisotropic response expected from continuum elasticity theory. In the last part of the thesis, we show that viral assembly and disassembly are activated processes controlled by nucleation barrier, which can be explained adapting classical nucleation theory (CNT). We focus on the case of spherical shells and confirm that the underlying assumptions of CNT are surprisingly good in characterizing the assembly of discrete shells, using the physical model introduced in the previous parts. Finally, we also unveil an interesting closure mechanism during the assembly of capsids.