  The nature of the cosmological dark matter remains elusive. Recent studies
have advocated the possibility that dark matter could be composed of
ultra-light, self-interacting bosons, forming a Bose-Einstein condensate in the
very early Universe. We consider models which are charged under a global
U(1)-symmetry such that the dark matter number is conserved. It can then be
described as a classical complex scalar field which evolves in an expanding
Universe. We present a brief review on the bounds on the model parameters from
cosmological and galactic observations, along with the properties of galactic
halos which result from such a dark matter candidate.
