  Hydrodynamics is nowadays understood as an effective field theory that
describes the dynamics of the long-wavelength and slow-time fluctuations of an
underlying microscopic theory. In this work we extend the relativistic
hydrodynamics to third order in the gradient expansion for neutral fluids in a
general curved spacetime of $d$ dimensions. We find 58 new transport
coefficients, 19 due to third-order scalar corrections and 39 due to tensorial
corrections. In the particular case of a conformal fluid, the number of new
transport coefficients is reduced to 19, all of them due to third-order
tensorial corrections. The dispersion relations of linear fluctuations in the
third-order relativistic hydrodynamics is obtained, both in the rest frame of
the fluid and in a general moving frame. As an application we obtain some of
the transport coefficients of a relativistic conformal fluid in three
dimensions by using the AdS/CFT correspondence. These transport coefficients
are extracted from the dispersion relations of the linear fluctuations. The
gravity dual of the fluctuations in this conformal fluid is described by the
gravitational perturbations of four-dimensional anti-de Sitter black branes,
which are solutions of the Einstein equations with a negative cosmological
constant. To find the hydrodynamic quasinormal modes (QNMs) of the scalar
sector we use the SUSY quantum mechanics of the gravitational perturbations of
four-dimensional black branes. Such a symmetry allows us to find the
wavefunction of the scalar (or sound) sector in the hydrodynamic limit directly
from the wavefunction of the vector (or shear) sector, which is usually easier
to be found because the perturbation wave equations for the vector sector are
much simpler than the ones for the scalar sector.
