  The study of transverse spin effects is part of the scientific program of
COMPASS, a fixed target experiment at the CERN SPS. COMPASS investigates the
transversity PDFs in semi-inclusive DIS, using a longitudinally polarized muon
beam of 160 GeV/c impinging on a transversely polarized target. From 2002 to
2004, data have been collected using a $^6$LiD target transversely polarized.
Transversity has been measured using different quark polarimeters: the
azimuthal distribution of single hadrons, the azimuthal dependence of the plane
containing hadron pairs, and the measurement of the transverse polarization of
baryons ($\Lambda$ hyperons). All the asymmetries have been found to be small,
and compatible with zero, a result which has been interpreted as a cancellation
between the u and d-quark contributions. In 2007 COMPASS has taken data using a
NH$_3$ polarized proton target which will give complementary information on
transverse spin effects.
