Collective flow, its anisotropies and its event-to-event fluctuations in relativistic heavy-ion collisions, and the extraction of the specific shear viscosity of quark-gluon plasma (QGP) from collective flow data collected in heavy-ion collision experiments at RHIC and LHC are reviewed. Specific emphasis is placed on the similarities between the Big Bang of our universe and the Little Bangs created in heavy-ion collisions.

We develop a complete and consistent description for the hadron spectra from heavy ion collisions in terms of a few collective variables, in particular temperature, longitudinal and transverse flow. To achieve a meaningful comparison with presently available data, we also include the resonance decays into our picture. To disentangle the influences of transverse flow and resonance decays in the m T -spectra, we analyse in detail the shape of the m T -spectra.

Using a hydrodynamic model, we predict the transverse momentum dependence of the spectra and the elliptic flow for different hadrons in Au+Au collisions at √ s = 130 A GeV. The dependence of the differential and pt-integrated elliptic flow on the hadron mass, equation of state and freeze-out temperature is studied both numerically and analytically.

We use ͑3ϩ1͒-dimensional hydrodynamics with exact longitudinal boost invariance to study the influence of collision centrality and initial energy density on the transverse flow pattern and the angular distributions of particles emitted near midrapidity in ultrarelativistic heavy-ion collisions. We concentrate on radial flow and the elliptic flow coefficient v 2 as functions of the impact parameter and collision energy. We demonstrate that the finally observed elliptic flow is established earlier in the collision than the observed radial flow and thus probes the equation of state at higher energy densities. We point out that a phase transition from hadronic matter to a color-deconfined quark-gluon plasma leads to nonmonotonic behavior in both beam energy and impact parameter dependences which, if observed, can be used to identify such a phase transition. Our calculations span collision energies from the Brookhaven AGS ͑Alternating Gradient Synchrotron͒ to beyond the LHC ͑Large Hadron Collider͒; the QGP phase transition signature is predicted between the lowest available SPS ͑CERN Super Proton Synchrotron͒ and the highest RHIC ͑Brookhaven Relativistic Heavy Ion Collider͒ energies. To optimize the chances for applicability of hydrodynamics we suggest studying the excitation function of flow anisotropies in central uranium-uranium collisions in the side-on-side collision geometry.