The quantum theory of stress is developed within the atoms in molecules (AIM) framework. The complete local stress field is introduced and integrated within atomic basins, and it is shown that the kinetic term gives rise to the atomic virial theorem. The role of the potential part of the stress field in the AIM theory is discussed, and its necessary consideration in order to define atomic pressures presented. These atomic pressures are shown to tend to the thermodynamic limit as the size of the system grows. A link between the AIM theory and the theory of electronic separability has also been found. A set of simple examples illustrates our results.