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 nonlinearity. By introducing the Madelung representation of the wave function, we formulate the dynamics of the system in terms of the continuity and hydrodynamic Euler equations. The nonrelativistic 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 nonlinearity, 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 briefly discussed. General relativistic configurations with quartic nonlinearity are studied numerically with both nonrelativistic 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 , maximum central density of the order of 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 are Bose-Einstein condensate stars. We discuss the connection of our results with previous boson star models based on scalar field theory.