Standard formulas for fluctuations in a grand ensemble are used (i) to obtain a new reciprocal relation between density and pressure fluctuations at critical points, including the transition temperature in a Bose-Einstein gas, (ii) to discuss the fluctuations in population of the superfluid in the Bose-Einstein theory of liquid helium, and (iii) the energy fluctuations in liquid helium and in Debye crystals. Except for a Fermi-Dirac electron gas, all relative fluctuations become infinite asT→0°K. The paper discusses briefly the significance of these results with regard to (a) the interpretation of second sound in liquid helium, and (b) the state of equilibrium near absolute zero.