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A set of measurements of azimuthal asymmetries in the production of pairs ofidentified hadrons in deep-inelastic scattering of muons on transverselypolarised $^6$LiD (deuteron) and NH$_3$ (proton) targets is presented. Allavailable data collected in the years 2002-2004 and 2007/2010 with the COMPASSspectrometer using a muon beam of 160 GeV/$c$ at the CERN SPS were analysed.The asymmetries provide access to the transversity distribution functions via afragmentation function that in principle may be independently obtained from$e^+e^-$ annihilation data. Results are presented, discussed and compared toexisting measurements as well as to model predictions. Asymmetries of$\pi^{+}\pi^{-}$ pairs measured with the proton target as a function of theBjorken scaling variable are sizeable in the range $x> 0.032$, indicatingnon-vanishing transversity distribution and di-hadron interferencefragmentation functions. As already pointed out by several authors, the smallasymmetries of $\pi^{+}\pi^{-}$ measured on the $^6$LiD target can beinterpreted as indication for a cancellation of $u$ and $d$-quark transversitydistributions.

Transverse-spin-dependent azimuthal asymmetries of pion and kaon pairs  produced in muon-proton and muon-deuteron semi-inclusive deep inelastic  scattering