  COMPASS is a high-energy physics experiment operating on the M2 beam line at
the SPS at CERN. Using high energy muon and hadron beams the experiment covers
broad range of physics aspects in the field of the hadron structure and
spectroscopy. One of the important objectives of the COMPASS experiment is the
exploration of transverse spin structure of the nucleon via study of spin
(in)dependent azimuthal asymmetries with semi-inclusive deep inelastic
scattering (SIDIS) processes and starting from 2014 also with Drell-Yan (DY)
reactions. 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. Giving access to the entire "twist-2"
set of transverse momentum dependent (TMD) parton distribution functions (PDFs)
and fragmentation functions (FFs) COMPASS data trigger constant theoretical
interest and are being widely used in phenomenological analyses and global data
fits. In particular, unique x-$Q^{2}$-z-pT multidimensional results for
transverse spin asymmetries recently obtained by COMPASS will serve as a direct
and unprecedented input for TMD $Q^{2}$-evolution related studies, one of the
hottest topics in the field of spin-physics. In addition, measurement of the
Sivers and all other azimuthal effects in polarized Drell-Yan at COMPASS will
reveal another side of the spin-puzzle providing a link between SIDIS and
Drell-Yan branches. This will be a unique possibility to test 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 obtained by the collaboration for multi-dimensional
transverse spin asymmetries and to the physics aspects of COMPASS polarized
Drell-Yan program.
