A general theory of the (Langevin) acoustic radiation force in three-dimensional fields in lossless fluids in Eulerian and Lagrangian coordinates is given. It is based on Brillouin’s approach of an acoustic radiation stress identified in Eulerian coordinates with the negative time average of the momentum flux density and in Lagrangian coordinates with the time average of the Piola–Kirchhoff–Boussinesq stress. It is shown in comparison to other statements from the literature that these stresses are not in general identical but that in the linear-acoustics approximation, the difference between the results in the two representations vanishes if the radiation force acting on an object entirely surrounded by the sound-propagating fluid is considered. It is just this situation which is of experimental relevance.