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Traditional spintronics relies on spin transport by charge carriers, such as electrons in semiconductor crystals. The challenges for the realization of long-range electron spin transport include rapid spin relaxation due to electron scattering. Scattering and, in turn, spin relaxation can be effectively suppressed in excitonic devices where the spin currents are carried by electrically neutral bosonic quasiparticles: excitons or exciton-polaritons. They can form coherent quantum liquids that carry spins over macroscopic distances. The price to pay is a finite lifetime of the bosonic spin carriers. We present the theory of exciton ballistic spin transport which may be applied to a range of systems supporting bosonic spin transport, in particular to indirect excitons in coupled quantum wells. We describe the effect of spin-orbit interaction for the electron and the hole on the exciton spin, account for the Zeeman effect induced by external magnetic fields and long-range and short-range exchange splittings of the exciton resonances. We also consider exciton transport in the nonlinear regime and discuss the definitions of the exciton spin current, polarization current, and spin conductivity.