  The relativistic treatment of spin is a fundamental subject which has an old
history. In various physical contexts it is necessary to separate the
relativistic total angular momentum into an orbital and spin contribution.
However, such decomposition is affected by ambiguities since one can always
redefine the orbital and spin part through the so-called pseudo-gauge
transformations. We analyze this problem in detail by discussing the most
common choices of energy-momentum and spin tensors with an emphasis on their
physical implications, and study the spin vector which is a pseudo-gauge
invariant operator. We review the angular momentum decomposition as a crucial
ingredient for the formulation of relativistic spin hydrodynamics and quantum
kinetic theory with a focus on relativistic nuclear collisions, where spin
physics has recently attracted significant attention. Furthermore, we point out
the connection between pseudo-gauge transformations and the different
definitions of the relativistic center of mass. Finally, we consider the
Einstein-Cartan theory, an extension of conventional general relativity, which
allows for a natural definition of the spin tensor.
