We study the role of electron-electron Coulomb interaction in clean graphene and determine the d.c. conductivity, the shear viscosity, and the diamagnetic response of this unique material. Key for an understanding of the Coulomb interaction is the fact that clean, undoped graphene is quantum critical with a marginally irrelevant "fine structure constant". Using standard crossover arguments combined with a quantum kinetic theory, we derive scaling laws, valid near this quantum critical point, that dictate the nontrivial magnetic and charge response of interacting graphene. The most dramatic consequence of this analysis is the anomalous collision-dominated, hydrodynamic transport: the d.c. conductivity of clean graphene is shown to diverge for decreasing temperature (as the square of the logarithm of T), making this material a quantum critical metal. The shear viscosity vanishes as the temperature goes to zero, with an viscosity - entropy density ratio that is close to the behavior expected for a perfect fluid.