  Exploration of transverse spin structure of the nucleon via study of the spin
(in)dependent azimuthal asymmetries in semi-inclusive deep inelastic scattering
(SIDIS) and Drell-Yan (DY) reactions is one of the main aspects of the broad
physics program of the COMPASS experiment (CERN, Switzerland). In past decade
COMPASS has collected a considerable amount of polarized deuteron and proton
SIDIS data, while recent 2014 and 2015 runs were dedicated to the Drell-Yan
measurements. Results on SIDIS azimuthal effects provided so far by COMPASS
play an important role in general understanding of the three-dimensional nature
of the nucleon. Giving access to the entire "twist-2" set of transverse
momentum dependent (TMD) parton distribution functions (PDFs) and fragmentation
functions (FFs) COMPASS data are being widely used in phenomenological analyses
and experimental data fits. Recent unique and first ever x-$Q^{2}$-z-pT
multidimensional results for transverse spin asymmetries obtained by COMPASS
serve as a direct and unprecedented input for one of the hottest topics in the
field of spin-physics: the TMD $Q^{2}$-evolution related studies. In addition,
extraction of the Sivers and all other azimuthal effects from first ever
polarized Drell-Yan data collected recently by COMPASS will reveal another side
of the spin-puzzle clarifying the link between SIDIS and Drell-Yan branches.
This will be a unique possibility to test predicted universality and
key-features of TMD PDFs using essentially the same experimental setup and
exploring the same kinematical domain. In this review main focus will be given
to the very recent results from COMPASS multi-dimensional analysis of
transverse spin asymmetries and to the physics aspects of COMPASS polarized
Drell-Yan program.
