We investigate the non-linear evolution of the matter power spectrum by using a large set of high-resolution N-body/hydrodynamic simulations. The linear matter power in the initial conditions is consistently modified to mimic the presence of warm dark matter (WDM) particles which induce a small-scale cut-off in the power as compared to standard cold dark matter scenarios. The impact of such thermal relics is examined at small scales k > 1 h Mpc−1, at redshifts of z < 5, which are particularly important for the next generation of Lyman α forest, weak lensing and galaxy clustering surveys. We measure the mass and redshift dependence of the WDM non-linear matter power and provide a fitting formula which is accurate at the ∼2 per cent level below z= 3 and for particle masses of mWDM≥ 0.5 keV. The role of baryonic physics on the WDM-induced suppression is also quantified. In particular, we examine the effects of cooling, star formation and feedback from strong galactic winds. Finally, we find that a modified version of the halo model describes the shape of the WDM suppressed power spectra better than halofit. In the case of weak lensing however, the latter works better than the former, since it is more accurate on the relevant, mid-range scales, albeit very inaccurate on the smallest scales (k > 10 h Mpc−1) of the matter power spectrum.