  We analyze the possibility that due to their superfluid properties some
compact astrophysical objects may contain a significant part of their matter in
the form of a Bose-Einstein condensate. To study the condensate we use the
Gross-Pitaevskii equation, with arbitrary non-linearity. By introducing the
Madelung representation of the wave function, we formulate the dynamics of the
system in terms of the continuity equation and of the hydrodynamic Euler
equations. The non-relativistic and Newtonian Bose-Einstein gravitational
condensate can be described as a gas, whose density and pressure are related by
a barotropic equation of state. In the case of a condensate with quartic
non-linearity, the equation of state is polytropic with index one. In the
framework of the Thomas-Fermi approximation the structure of the Newtonian
gravitational condensate is described by the Lane-Emden equation, which can be
exactly solved. The case of the rotating condensate is also discussed. General
relativistic configurations with quartic non-linearity are studied numerically
with both non-relativistic and relativistic equations of state, and the maximum
mass of the stable configuration is determined. Condensates with particle
masses of the order of two neutron masses (Cooper pair) and scattering length
of the order of 10-20 fm have maximum masses of the order of 2 M_sun, maximum
central density of the order of 0.1-0.3 10^16 g/cm^3 and minimum radii in the
range of 10-20 km. In this way we obtain a large class of stable astrophysical
objects, whose basic astrophysical parameters (mass and radius) sensitively
depend on the mass of the condensed particle, and on the scattering length. We
also propose that the recently observed neutron stars with masses in the range
of 2-2.4 M_sun are Bose-Einstein Condensate stars.
