The COMPASS experiment at CERN studies the spin structure of the nucleon, using its unique polarized target in both longitudinal and transverse polarization modes. The future Drell-Yan measurement by the COMPASS collaboration proposes to access the transverse momentum dependent parton distribution functions, namely Sivers and Boer-Mulders functions. The Drell-Yan process with unpolarized and with transversely polarized target is a very promising tool for this purpose, complementary to the semi-inclusive deep inelastic scattering measurements available from COMPASS, HERMES and JLab experiments. Also interesting related studies like the J/ψ-Drell-Yan duality are proposed. The physics case, as well as the most important technical aspects of this project are presented.