COMPASS is a fixed target experiment at the CERN SPS M2 beam line, taking data since year 2002. Part of the physics programme is dedicated to the study of the transverse spin and transverse momentum structure of the nucleon, by means of semi-inclusive deeply inelastic lepton scattering (SIDIS). For these measurements, data have been collected using transversely polarised proton (NH3) and deuteron (6LiD) targets. A selection of the results on the azimuthal asymmetries measured is presented, with particular focus on the most recent measurements from the data collected in 2007 and 2010 with the proton target. These results exhibit clear signals for the Collins asymmetry, interpreted as a convolution of a non-zero transversity PDF and Collins fragmentation function, and for the Sivers asymmetry which is related to the Sivers function, the most famous and discussed of the TMD PDFs. The data collected with the 6LiD target, have also been recently analysed to search for the azimuthal modulations in the production of hadrons which are expected to be present in the unpolarised part of the SIDIS cross-section. The azimuthal hadron asymmetries, which are related to the Boer-Mulders TMD PDF, show strong and somewhat puzzling kinematical dependencies.
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