  Gamma-ray bursts and their afterglows are thought to be produced by an
ultra-relativistic jet. One of the most important open questions is the outflow
composition: the energy may be carried out from the central source either as
kinetic energy (of baryons and/or pairs), or in electromagnetic form (Poynting
flux). While the total observable flux may be indistinguishable in both cases,
its polarization properties are expected to differ markedly. The prompt
emission and afterglow polarization are also a powerful diagnostic of the jet
geometry. Again, with subtle and hardly detectable differences in the output
flux, we have distinct polarization predictions. In this review we briefly
describe the theoretical scenarios that have been developed following the
observations, and the now large observational datasets that for the prompt and
the afterglow phases are available. Possible implications of polarimetric
measurements for quantum gravity theory testing are discussed, and future
perspectives for the field briefly mentioned.
