COMPASS is a fixed-target experiment operating on north area of SPS (M2 beamline) at CERN. An important part of the physics programme of the experiment is the exploration of the transverse spin structure of the nucleon via measurements of spin-(in)dependent azimuthal asymmetries in the semi-inclusive DIS and, recently, also in Drell-Yan processes. Drell-Yan measurements with a 190 GeV/c $\pi^-$ beam impinging on a transversely polarized NH3 target started in the year 2015 (18 weeks data taking) and will be continued in 2018. The measurement of the Sivers and other azimuthal asymmetries in polarized SIDIS and Drell-Yan performed by COMPASS provides a unique possibility to test (pseudo-)universal features of transverse momentum-dependent part...