  We review the present state of knowledge regarding the transverse
polarisation (or transversity) distributions of quarks. After some generalities
on transverse polarisation, we formally define the transversity distributions
within the framework of a classification of all leading-twist distribution
functions. We describe the QCD evolution of transversity at leading and
next-to-leading order. A comprehensive treatment of non-perturbative
calculations of transversity distributions (within the framework of quark
models, lattice QCD and QCD sum rules) is presented. The phenomenology of
transversity (in particular, in Drell-Yan processes and semi-inclusive
leptoproduction) is discussed in some detail. Finally, the prospects for future
measurements are outlined.
