  In this short review, which is based on the works published between 2011 and
2016, we discuss the problem of the propagation of a collective orbital
excitation (orbiton) created in the Mott insulating and antiferromagnetic
ground state. On the theoretical side, the problem is solved by mapping a
Kugel-Khomskii spin-orbital model describing an orbiton moving in an
antiferromagnet onto an effective t-J model with a 'single hole' moving in an
antiferromagnet. The most important consequence of the existence of the above
mapping is the fractionalisation of the electron's spin and orbital degree of
freedom in the 1D antiferromagnets---a spin-orbital separation phenomenon that
is similar to the spin-charge separation in 1D but corresponds to an exotic
regime where spinons are faster than holons. Besides a detailed explanation and
benchmarking of the mapping, in this review we also discuss its application to
several relatively realistic spin-orbital models, that are able to describe the
experimentally observed orbital excitation spectra of copper and iridium
oxides.
