  A new approach is described to help improve the foundations of relativistic
viscous fluid dynamics and its coupling to general relativity. Focusing on
neutral conformal fluids constructed solely in terms of hydrodynamic variables,
we derive the most general viscous energy-momentum tensor yielding equations of
motion of second order in the derivatives, which is shown to provide a novel
type of generalization of the relativistic Navier-Stokes equations for which
causality holds. We show how this energy-momentum tensor may be derived from
conformal kinetic theory. We rigorously prove existence, uniqueness, and
causality of solutions of this theory (in the full nonlinear regime) both in a
Minkowski background and also when the fluid is dynamically coupled to
Einstein's equations. Linearized disturbances around equilibrium in Minkowski
spacetime are stable in this causal theory. A numerical study reveals the
presence of an out-of-equilibrium hydrodynamic attractor for a rapidly
expanding fluid. Further properties are also studied and a brief discussion of
how this approach can be generalized to non-conformal fluids is presented.
