  We develop a geometric formulation of fluid dynamics, valid on arbitrary
Riemannian manifolds, that regards the momentum-flux and stress tensors as
1-form valued 2-forms, and their divergence as a covariant exterior derivative.
We review the necessary tools of differential geometry and obtain the
corresponding coordinate-free form of the equations of motion for a variety of
inviscid fluid models -- compressible and incompressible Euler equations,
Lagrangian-averaged Euler-$\alpha$ equations, magnetohydrodynamics and
shallow-water models -- using a variational derivation which automatically
yields a symmetric momentum flux. We also consider dissipative effects and
discuss the geometric form of the Navier--Stokes equations for viscous fluids
and of the Oldroyd-B model for visco-elastic fluids.
