  Various sum rules accounting for the coupling between density and particle
excitations and emphasizing in an explicit way the role of the Bose-Einstein
condensation are discussed. Important consequences on the fluctuations of the
particle operator as well as on the structure of elementary excitations are
reviewed. These include a recent generalization of the Hohenberg-Mermin-Wagner
theorem holding at zero temperature. (To appear in "Bose-Einstein
Condensation", A.Griffin, D.Snoke and S.Stringari eds., Cambridge Univ. Press)
