  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.
