Recent observational studies of remote supernovae have suggested the existence of cosmic vacuum with the density which is larger than the total density of all the other forms of cosmic energy in the Universe. Vacuum produces the field of anti-gravity and induces acceleration of the cosmological expansion. It is this acceleration of the expansion that has been discovered in the observations. The discovery of the cosmic vacuum leads to a drastic change in the current concepts concerning the present state of the Universe. It poses also a number of new important problems in both cosmology and fundamental physics. Why is the density of vacuum as that as it has been found in observations? Why are the components of the cosmic medium have different, but close to each other, on the order of magnitude, values of their densities? On the other hand, this discovery made at the large cosmological distances (hundreds and thousands megaparsecs) provides new insights into the dynamics of the nearby area in the Universe, the motions of galaxies in the local volume with the radius of 10-20 megaparsec where the cosmological expansion was first discovered.