  Recent progress in the formulation of relativistic hydrodynamics for
particles with spin one-half is reviewed. We start with general arguments
advising introduction of a tensor spin chemical potential that plays a role of
the Lagrange multiplier coupled to the spin angular momentum. Then, we turn to
a discussion of spin-dependent distribution functions that have been recently
proposed to construct a hydrodynamic framework including spin and serve as a
tool in phenomenological studies of hadron polarization. Distribution functions
of this type are subsequently used to construct the equilibrium Wigner
functions that are employed in the semi-classical kinetic equation. The
semi-classical expansion elucidates several aspects of the hydrodynamic
approach, in particular, shows the ways in which different possible versions of
hydrodynamics with spin can be connected by pseudo-gauge transformations. These
results point out at using the de~Groot - van~Leeuwen - van~Weert versions of
the energy-momentum and spin tensors as the most natural and complete physical
variables. Finally, a totally new method is proposed to design hydrodynamics
with spin, which is based on the classical treatment of spin degrees of
freedom. Interestingly, for small values of the spin chemical potential the new
scheme brings the results that coincide with those obtained before. The
classical approach also helps us to resolve problems connected with the
normalization of the spin polarization three-vector. In addition, it clarifies
the role of the Pauli-Lubanski vector and the entropy current conservation. We
close our review with several general comments presenting possible future
developments of the discussed frameworks.
