  In the history of cosmology physical paradoxes played important role for
development of contemporary world models. Within the modern standard
cosmological model there are both observational and conceptual cosmological
paradoxes which stimulate to search their solution. Confrontation of
theoretical predictions of the standard cosmological model with the latest
astrophysical observational data is considered. A review of conceptual problems
of the Friedmann space expending models, which are in the bases of modern
cosmological model, is discussed. The main paradoxes, which are discussed in
modern literature, are the Newtonian character of the exact Friedmann equation,
the violation of the energy conservation within any comoving local volume,
violation of the limiting recession velocity of galaxies for the observed high
redshift objects. Possible observational tests of the nature of the
cosmological redshift are discussed
