Context : The relative abundance of the dust grain types in the interstellar medium is directly linked to physical quantities that trace the evolution of galaxies .
Because of the poor spatial resolution of the infrared/submillimetre data , we are able to study the dependence of the resolved infrared Spectral Energy Distribution ( SED ) across regions of the interstellar medium ( ISM ) with different physical properties in just a few objects .

Aims : We study the dust properties of the whole disc of M 33 at spatial scales of \sim 170 pc .
This analysis allows us to infer how the relative dust grain abundance changes with the conditions of the ISM , study the existence of a submillimetre excess and look for trends of the gas-to-dust mass ratio ( GDR ) with other physical properties of the galaxy .

Methods : For each pixel in the disc of M 33 we fit the infrared SED using a physically motivated dust model that assumes an emissivity index \beta close to 2 .
We apply a Bayesian statistical method to fit the individual SEDs and derive the best output values from the study of the probability density function of each parameter .
We derive the relative amount of the different dust grains in the model , the total dust mass , and the strength of the interstellar radiation field ( ISRF ) heating the dust at each spatial location .

Results : The relative abundance of very small grains ( VSGs ) tends to increase , and for big grains ( BGs ) to decrease , at high values of H \alpha luminosity .
This shows that the dust grains are modified inside the star-forming regions , in agreement with a theoretical framework of dust evolution under different physical conditions .
The radial dependence of the GDR is consistent with the shallow metallicity gradient observed in this galaxy .
The strength of the ISRF derived in our model correlates with the star formation rate ( SFR ) in the galaxy in a pixel by pixel basis .
Although this is expected it is the first time that a correlation between both quantities is reported .
We produce a map of submillimetre excess in the 500 \mu m SPIRE band for the disc of M 33 .
The excess can be as high as 50 % and increases at large galactocentric distances .
We further study the relation of the excess with other physical properties of the galaxy and find that the excess is prominent in zones of diffuse ISM outside the main star-forming regions , where the molecular gas and dust surface density are low .

Conclusions :