  One of the important objectives of the COMPASS experiment (CERN, SPS north
area) is the exploration of transverse spin structure of nucleon via study of
spin (in)dependent azimuthal asymmetries with semi-inclusive deep inelastic
scattering (SIDIS) processes and recently also with Drell-Yan (DY) reactions.
In the past twelve years series of measurements were made in COMPASS, using 160
GeV/c longitudinally polarized muon beam and polarized $^6LiD$ and $NH_3$
targets. Drell-Yan measurements with high energy (190 GeV/c) pion beam and
transversely polarized $NH_3$ target started in 2014 with a pilot-run have been
followed by 140 days of data taking in 2015. 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. In addition, 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 is given to the very recent results obtained by the COMPASS collaboration
from first ever multi-dimensional extraction of SIDIS transverse spin
asymmetries.
