[en] The aim of this introduction, which is far from exhaustive, was to give an overview on the richness of transverse spin quantity and its differences in comparison with helicity. From the experimental point of view, the physics of quark transversity in deep inelastic reaction is still practically unexplored. This situation will certainly change rapidly, with planned experiments at DESY (HERMES), Brookhaven (RHIC) and CERN (COMPAS), but there is a long way before knowing the transversity distribution, δq(x), as precisely as the helicity distribution, Δq(x), now. Unless polarized anti-proton beams become feasible, experiments probing quark transversity will rely mainly on 'quark polarimeters', like Λ's or the Collins effect. These polarimeters will have to be calibrated at e⁺e^- colliders. The Collins polarimeter will by the way allow the flavor decomposition of δq(x), using mesons of various charging and strangeness. Quark polarimetry is by itself an interesting topic of non-perturbative QCD, and may teach us something about the breaking of chiral symmetry. Let us recall that, if chiral symmetry were unbroken, transversity would be undefined. The transversity physics program is not at all a 'remake' of the helicity one. Helicity and transversity probe rather different aspects of the hadron structure. Differences between Δq(x) and δq(x) will reveal non-relativistic effects in the baryon wave function. Also δq(x) does not couples to gluon distributions, thus it is free from anomaly. In that respect it is a more clean probe than Δq(x). In fact, the combination of helicity and transversity measurements will perhaps be the most interesting. Polarized parton densities taking only the helicity degree of freedom are almost 'classical'. Quantum aspects of spin correlations, like violation of Bell's inequality, can be found only when varying the spin quantification axis. Classical-like densities in one basis appear as quantum interferences in the other basis. Transversely polarized experiment may also be a tool to detect new physics. On the theoretical side, much work has been done, but much remains also to be done. Which reggeon, if any, governs δq(x) at low x has not been discussed, to our knowledge. The connection between the tensor charge of the baryons and their magnetic and electric-dipole moments has to be clarified. The contents of the paper is as fallows: 1. Pre-history; 2. Transversity versus helicity; 3. The massless limit; 4. Transversity distribution δq(x). The diquark spectator model; 5. Soffer inequality; 6. Tensor charge sum rule; 7. t-channel analysis; 8. Selection rules for δq(x) measurements; 9. Evolution with Q²; 10. Quark polarimetry. The sheared jet (Collins) effect; 11. Single-spin asymmetries in inclusive experiments; 12. Quark distribution dependent on both spin and k_T; 13. First evidence of quark transversity. (author)