The tensor virial equations for an aggregate of particles are developed. They are tensor equations of second rank, and on contraction yield the familiar virial equation. To illustrate their application, the diffusion of molecules through a gas is considered and a kinetic derivation of the Navier-Stokes equations is worked out. The latter may here be carried out in more general terms than is usual, the generalization leading to results that are of interest in turbulence theory. In anisotropic systems the use of the tensor in place of the scalar virial becomes imperative. An astrophysical example of this (anisotropic star cluster) is given in outline.