I will review some results of observational cosmology which critically cast doubt upon the foundations of the standard cosmology: 1) The redshifts of the galaxies are due to the expansion of the Universe; 2) The cosmic microwave background radiation and its anisotropies come from the high energy primordial Universe; 3) The abundance pattern of the light elements is to be explained in terms of the primordial nucleosynthesis; 4) The formation and evolution of galaxies can only be explained in terms of gravitation in the cold dark matter theory of an expanding Universe. The review does not pretend to argue against this standard scenario in favour of an alternative theory, but to claim that cosmology is still a very young science and should leave the door wide open to other positions.