  We derive constraints on elastic scattering between baryons and dark matter
using the cosmic microwave background (CMB) data from the Planck satellite and
the Lyman-alpha forest data from the Sloan Digital Sky Survey. Elastic
scattering allows baryons and dark matter to exchange momentum, affecting the
dynamics of linear density perturbations in the early Universe. We derive
constraints to scattering cross sections of the form sigma \propto v^n,
allowing for a wide range of velocity dependencies with n between -4 and 2. We
improve and correct previous estimates where they exist, including
velocity-independent cross section as well as dark matter millicharge and
electromagnetic dipole moments. Lyman-alpha forest data dominates the
constraints for n>-3, where the improvement over CMB data alone can be several
orders of magnitude. Dark matter-baryon scattering cannot affect the halo mass
function on mass scales M>10^{12} M_{solar}. Our results imply,
model-independently, that a baryon in the halo of a galaxy like our own Milky
Way, does not scatter from dark matter particles during the age of the galaxy.
