We present conditional existence results for the Landau equation with Coulomb potential. Despite lack of a comparison principle for the equation, the proof of existence relies on barrier arguments and parabolic regularity theory. The Landau equation arises in kinetic theory of plasma physics. It was derived by Landau and serves as a formal approximation to the Boltzmann equation when grazing collisions are predominant. We also present long-time existence results for the isotropic version of the Landau equation with Coulomb potential.