  We consider the hydrodynamics of relativistic conformal field theories at
finite temperature and its slow motions limit, where it reduces to the
incompressible Navier-Stokes equations. The symmetries of the equations and
their solutions are analyzed. We construct exact solutions with finite time
singularities of one-dimensional relativistic conformal hydrodynamic motions,
and use them to generate multi-dimensional solutions via special conformal
transformations. These solutions, however, are shown to have no non-trivial
slow motions limit. A simple non-equilibrium steady state in the form of a
shock solution is constructed, and its inner structure is analyzed. We
demonstrate that the derivation of the gravitational dual description of
conformal hydrodynamics is analogous to the derivation of hydrodynamics
equations from the Boltzmann equation. The shock solution is shown to
correspond to a domain-wall solution in gravity. We show that the solutions to
the non-relativistic incompressible Navier-Stokes equations play a special role
in the construction of global solutions to gravity.
