Within hydrodynamics we study the effects of the initial spatial anisotropy in non-central heavy-ion collisions on the momentum distributions of the emitted hadrons. We show that the elliptic flow measured at midrapidity in 158 A GeV/c Pb+Pb collisions can be quantitatively reproduced by hydrodynamic expansion, indicating early thermalization in the collision. We predict the excitation functions of the 2 nd and 4 th harmonic flow coefficients from AGS to LHC energies and discuss their sensitivity to the quark-hadron phase transition.The recent observation of transverse collective flow phenomena in non-central heavy-ion collisions at ultrarelativistic beam energies [1][2][3][4] has led to renewed intense theoretical interest in this topic (see [5] for a review). Collective flow is the consequence of pressure in the system and thereby provides access to the equation of state of the hot and dense matter ("fireball") formed in the reaction zone. This access is indirect since the flow in the final state represents a time integral over the pressure history of the fireball. Sorge [6] has argued that different types of transverse flow (radial, directed, elliptic, see [5]) show different sensitivities to the early and late stages of the collision such that a combination of flow observables may allow for a more differential investigation of the equation of state. In particular, he pointed out that the elliptic flow (which develops in non-central collisions predominantly at midrapidity and manifests itself as an elliptic deformation of the hadronic momentum distributions around the beam axis [7,8]) is a signature for the early stage of the collision: its driving force is the spatial eccentricity of the dense nuclear overlap region which, if thermalized quickly enough, leads to an anisotropy of the pressure gradients which cause the expansion. Since the developing anisotropic flow reduces the eccentricity of the fireball, it acts against its own cause and thus shuts itself off after some time. Radial flow, on the other hand, responds to the absolute magnitude of the pressure gradients and not only to their anisotropy; it therefore exists also in central collisions, and in non-central collisions it continues to grow even after the initial elliptic spatial deformation of the fireball has disappeared.A phase transition from a hadron gas to a colordeconfined quark-gluon plasma causes a softening of the * On leave of absence from Institut für Theoretische Physik, Universität Regensburg. Email: Ulrich.Heinz@cern.ch equation of state: as the temperature crosses the critical value for the phase transition, the energy and entropy densities increase rapidly while the pressure rises slowly. The resulting small ratio of p/e at the upper end of the transition region ("the softest point" [9]) weakens the build-up of flow as the system passes through it. Shuryak [10] and van Hove [11] therefore suggested that a plot of the mean transverse momentum against the central multiplicity density should show a plateau. Later hydrodynamic calculati...