High electron mobility in graphene is one of its most interesting properties for potential applications. Despite intensive efforts, both experimental and theoretical, we still have no complete understanding of main electron scattering mechanisms and main limiting factors restricting the mobility. It is clear that a long-range scattering is important since short-range scatterers with radius of potential smaller than the electron wavelength are irrelevant for massless Dirac fermions. Three most probable candidates are charge impurities, scattering by elastic deformations created by frozen ripples, and resonant scattering centers (the last case also deals with long-range effects due to divergence of the scattering length). I review a theory of these mechanisms, together with relevant experimental results and first-principle calculations. It seems that the resonant scattering is the best candidate but the issue requires further research. I discuss also peculiarities of electron transport in bilayer graphene and temperature dependence of resistivity for freely suspended graphene samples. In the latter case, two-phonon processes involving bending mode give probably the main contribution.