  COMPASS is a high-energy physics experiment operating at the SPS at CERN.
Wide physics program of the experiment comprises study of hadron structure and
spectroscopy with high energy muon and hadrons beams. As for the muon-program,
one of the important objectives of the COMPASS experiment is the exploration of
the transverse spin structure of the nucleon via spin (in)dependent azimuthal
asymmetries in single-hadron production in deep inelastic scattering of
polarized leptons off transversely polarized target. For this purpose a series
of measurements were made in COMPASS, using 160 GeV/c longitudinally polarized
muon beam and transversely polarized $^{6}LiD$ (in 2002, 2003 and 2004) and
$NH_{3}$ (in 2007 and 2010) targets. The experimental results obtained by
COMPASS for unpolarized target azimuthal asymmetries, Sivers and Collins
effects and other azimuthal observables play an important role in the general
understanding of the three-dimensional nature of the nucleon. Giving access to
the entire "twsit-2" set of transverse momentum dependent parton distribution
functions and fragmentation functions COMPASS data triggers constant
theoretical interest and is being widely used in phenomenological analyses and
global data fits. In this review main focus is given to the very recent results
obtained by the COMPASS collaboration from first ever multi-dimensional
extraction of transverse spin asymmetries.
