Recent observations at low and high redshift seem to confirm the presence of dust in the InterGalactic Medium (IGM) and especially in the IntraCluster Medium (ICM), at very low abundances. We have studied the impact of this dust on the IGM conditions, in terms of heating and cooling. Firstly we have described the dust emission by an analytical model, and we have proved that the dust IR emission can be considered as the dominant cooling agent of the ICM, when the temperature is greater than T=107 K and the dust-to-gas mass ratio exceeds a value of Z_d = 2× 10-5. This cooling process is more efficient for small grains, and fills the gap between the cooling by free-free emission at T > 108 K and other cooling processes such as recombination, collisional ionisation and collisional excitation at T < 105 K. Secondly, when the UV Background is strong enough, JUV(ν) ≥10-22 (ν/(13.6 eV))-1 erg s-1 cm-2 sr-1 Hz-1, and the temperature is low enough, Te ≤105 K, dust grains become an efficient heating agent of the IGM. Moreover these two opposite processes due to dust have opposite impact on the IGM at different scales: the cooling should be dominant at large scale (in the ICM), whereas the heating is efficient at small scale (near quasars or star formation regions). The issue of the dust replenishment of the IGM is also considered, taking into account the process of galactic dust ejection and intergalactic dust production. The recent observations of an intergalactic star population and the estimate of the reionisation (pop III stars) at z=12-20, both point toward dust pollution of the IGM early in the cosmic history, with possible important consequences for the evolution scenarii of the Universe.