  We present the first generalization of Navier-Stokes theory to relativity
that satisfies all of the following properties: (a) the system coupled to
Einstein's equations is causal and strongly hyperbolic; (b) equilibrium states
are stable; (c) all leading dissipative contributions are present, i.e., shear
viscosity, bulk viscosity, and thermal conductivity; (d) non-zero baryon number
is included; (e) entropy production is non-negative in the regime of validity
of the theory; (f) all of the above holds in the nonlinear regime without any
simplifying symmetry assumptions. These properties are accomplished using a
generalization of Eckart's theory containing only the hydrodynamic variables,
so that no new extended degrees of freedom are needed as in
M\"uller-Israel-Stewart theories. Property (b), in particular, follows from a
more general result that we also establish, namely, sufficient conditions that
when added to stability in the fluid's rest frame imply stability in any
reference frame obtained via a Lorentz transformation. All our results are
mathematically rigorously established. The framework presented here provides
the starting point for systematic investigations of general-relativistic
viscous phenomena in neutron star mergers.
