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The influence of retardation on the energy of interaction between two neutral atoms is investigated by means of quantum electrodynamics. As a preliminary step, Part I contains a discussion of the interaction between a neutral atom and a perfectly conducting plane, and it is found that the influence of retardation leads to a reduction of the interaction energy by a correction factor which decreases monotonically with increasing distance $R$. This factor is equal to unity for $R$ small compared with the wave-lengths corresponding to the atomic frequencies, and is proportional to ${R}^{\ensuremath{-}1}$ for distances large compared with these wave-lengths. In the latter case the total interaction energy is given by $\ensuremath{-}\frac{3\ensuremath{\hbar}c\ensuremath{\alpha}}{8\ensuremath{\pi}{R}^{4}}$, where $\ensuremath{\alpha}$ is the static polarizability of the atom. Although the problem of the interaction of two atoms discussed in Part II is much more difficult to handle mathematically, the results are very similar. Again the influence of retardation can be described by a monotonically decreasing correction factor which is equal to unity for small distances and proportional to ${R}^{\ensuremath{-}1}$ for large distances. In the latter case the energy of interaction is found to be $\ensuremath{-}\frac{23\ensuremath{\hbar}c{\ensuremath{\alpha}}_{1}{\ensuremath{\alpha}}_{2}}{4\ensuremath{\pi}{R}^{7}}$.