We present a new study on the elastic scattering cross section of dark matter (DM) and neutrinos using the latest cosmological data from Planck and large-scale structure experiments. We find that the strongest constraints are set by the Lyman-a forest, giving aDM v < 10(-33) (172Dm/GeV) CM2 if the cross section is constant and a present-day value of aDM v < 10(-45) (172Dm/GeV) CM2 if it scales as the temperature squared. These are the most robust limits on DM-neutrino interactions to date, demonstrating that one can use the distribution of matter in the Universe to probe dark ("invisible") interactions. Additionally, we show that scenarios involving thermal MeV DM and a constant elastic scattering cross section naturally predict (i) a cut-off in the matter power spectrum at the Lyman-a scale, (ii) Neff 3.5 0.4, (iii) Ho 71 3 km s-1 Mpc-1 and (iv) the possible generation of neutrino masses.