The paradox of global universe rotation, as it seems, is today one of its greatest mysteries of nature. The idea originated less than a hundred years, and getting the most correct answer was possible only after the creation of modern cosmology. In this book, the authors forward their visions on the universal rotation problem. At the same time, the arguments in favor of its global rotation absence are presented as well. In fact, different theoretical and observational aspects of evidence that supports the possible rotation of the universe on different cosmological scales were investigated. It was shown that there are correlations between angular momentum and size of the structures. The presented observational picture is that the galaxies, their pairs and compact groups have a non vanishing angular momentum. Moreover, these momenta have the definite tendency to their alignment. An analysis of the distribution of position angles of more than ten thousand extended radio sources shows that the spatial orientation of axes of these objects is anisotropic: they are mostly oriented not in the direction of the celestial pole, but rather in the equatorial direction. The probability that the sky distribution of axes is isotropic is less than 0.00004. It was argued that our universe rotates differentially. Based on the spinning part of Papapetrous equations, it was shown that the rotation radically depends on the properties of the epoch of universal growth. The universes angular velocities have been calculated for three main cosmological epochs: the matter dominated epoch, the transient from matter to vacuum dominated epoch, and the vacuum dominated epoch. In the framework of general relativity, the nonstationary Bianchi type VIII cosmological models with rotation were proposed. As the sources of gravity have been chosen, they are known as comoving perfect fluid with non-comoving dust, comoving perfect fluid with pure radiation, and comoving anisotropic fluid. All of the models contain a rotating dark energy. Recent observations, such as the anomalies in the temperature angular distribution of the cosmic microwave background (CMB), indicate a preferred direction in the universe. The foundation of modern cosmology, however, relies on a homogeneous and isotropic distribution of matter on large scales. Here, the authors consider the preferred axis in the CMB parity violation. The authors also found that this axis coincides with the preferred axes of the CMB quadrupole and octopole, and they all align with the direction of the CMB kinematic dipole, which does not have a cosmological origin. The several exact non-stationary Goedel-type solutions that belong to a large class of shear-free spatially homogeneous spacetimes were presented. The observational effects of the universal rotation in cosmology are analyzed. It was shown that the pure cosmic rotation does not produce either causality violations, parallax effects, or anisotropy of the microwave background radiation. A possible way to detect the cosmic rotation is to search for the angular dependencies of the standard cosmological tests, and a number of new observations can be proposed.
