Three distinct forms are derived for the force virial contribution to the pressure and stress tensor of a collection of atoms interacting under periodic boundary conditions. All three forms are written in terms of forces acting on atoms, and so are valid for arbitrary many-body interatomic potentials. All three forms are mathematically equivalent. In the special case of atoms interacting with pair potentials, they reduce to previously published forms. (i) The atom-cell form is similar to the standard expression for the virial for a finite nonperiodic system, but with an explicit correction for interactions with periodic images. (ii) The atom form is particularly suited to implementation in modern molecular dynamics simulation codes using spatial decomposition parallel algorithms. (iii) The group form of the virial allows the contributions to the virial to be assigned to individual atoms.