New results on transverse spin asymmetries from COMPASS

One of the important objectives of the COMPASS experiment is the exploration of transverse spin structure of nucleon via spin (in)dependent azimuthal asymmetries in semi-inclusive deep inelastic scattering (SIDIS) of polarized leptons (and soon also Drell-Yan (DY) reactions with $\pi^-$) off transversely polarized target. For this purpose a series of measurements were made in COMPASS, using 160 GeV/c longitudinally polarized muon beam and polarized $^6LiD$ and $NH_3$ targets and are foreseen with 190 GeV/c $\pi^-$ beam on polarized $NH_3$. The experimental results obtained by COMPASS for azimuthal effects in SIDIS play an important role in the general understanding of the three-dimensional nature of the nucleon and are widely used in theoretical analyses and global data fits. Future first ever polarized DY-data from COMPASS compared with SIDIS results will open a new chapter probing general principles of QCD TMD-formalism. In this review main focus will be given to the very recent COMPASS results obtained for SIDIS transverse spin asymmetries from four "Drell-Yan" $Q^2$-ranges.