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.

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We present next-to-leading order predictions for double transverse-spin asymmetries in Drell-Yan dilepton production initiated by proton-antiproton scattering. The kinematic region of the proposed PAX experiment at GSI: 30 s 200 GeV 2 and 2 M 7 GeV is examined. The Drell-Yan asymmetries turn out to be large, in the range 20-40%. Measuring these asymmetries would provide the cleanest determination of the quark transversity distributions.

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