We study the effects of disorder in the electronic properties of graphene multilayers, with special focus on the bilayer and the infinite stack. At low energies and long wavelengths, the electronic self-energies and density of states exhibit behavior with divergences near half filling. As a consequence, the spectral functions and the conductivities acquire anomalous properties. In particular, we show that the quasiparticle decay rate has a minimum as a function of energy, there is a universal minimum value for the in-plane conductivity of order e(2)/h per plane and, unexpectedly, the c-axis conductivity is enhanced by disorder at low doping, leading to an enormous conductivity anisotropy at low temperatures.