  We review the phenomenon of equilibrium fluctuations in the number of
condensed atoms in a trap containing N atoms total. We start with a history of
the Bose-Einstein distribution, the Einstein-Uhlenbeck debate concerning the
rounding of the mean number of condensed atoms near a critical temperature, and
a discussion of the relations between statistics of BEC fluctuations in the
grand canonical, canonical, and microcanonical ensembles.
  Next we discuss different approaches capable of providing approximate
analytical results and physical insight into the problem of fluctuations. In
particular, we describe the master equation (similar to the quantum theory of
the laser) and canonical-ensemble quasiparticle approaches which give the most
accurate and physically transparent picture of the BEC fluctuations.
  In the last part we describe condensate fluctuations in the interacting Bose
gas. In particular, we show that the canonical-ensemble quasiparticle approach
works very well for the interacting gases and yields analytical formulas for
the characteristic function and all moments of the condensate fluctuations. In
most cases the ground-state occupation fluctuations are anomalously large and
are not Gaussian even in the thermodynamic limit. We clarify a crossover
between the ideal and weakly-interacting-gas statistics which is governed by a
pair-correlation squeezing mechanism.
