The diffuse interstellar medium (ISM) hosts the first steps of interstellar chemistry and the seeds of dense structures. Since its turbulent pressure by far exceeds its thermal pressure, turbulence must play a prominent role in its evolution. Fed at galactic scales, turbulent energy cascades down to the dissipation scales, but as in both laboratory and atmospheric turbulence, it does so in an intermittent way : only a tiny fraction of the small-scales is fed by the turbulent cascade, so that dissipation occurs in bursts. In diffuse molecular clouds, where they can be observed, the signatures of intermittency are: (1) the non-Gaussian statistics of velocity increments, and (2) the existence of coherent structures of intense velocity-shear that appear to channel the large-scale turbulent energy down to milliparsec scales. Attempts at modelling the warm chemistry triggered in the diffuse ISM by bursts of turbulent dissipation are promising : in this framework, the so far unexplained molecular richness observed in this medium is naturally understood, in particular its CH+, HCO+ and CO abundances. Turbulent dissipation is also likely at the origin of the H2 rotational line emission of the diffuse ISM and of a significant fraction of its [C II] emission.