Electron properties of graphene are described in terms of the Dirac fermions. Here we thoroughly outline the elastic scattering theory for the two-dimensional massive Dirac fermions in the presence of an axially symmetric potential smooth on the lattice scale. While the massless limit is relevant for pristine graphene, keeping finite mass allows for generalizations onto situations with broken symmetry between the two sublattices, and provides a link to the scattering theory of electrons in a parabolic band. We demonstrate that the Dirac theory breaks down for the potentials whose short-distance behavior is more singular than 1/r. The formalism is then applied to scattering off a short-ranged potential whose size is smaller than the wavelength. Next we consider the Coulomb potential scattering, where the Dirac theory in two dimensions is consistent for a point scatterer only for the effective ``fine structure constant'' below 1/2. From the scattering phase shifts we obtain the exact Coulomb transport cross-section in terms of the fine-structure constant. The results are relevant for transport in graphene in the presence of potential impurities that do not induce scattering between the Dirac points.