Graphene is one atom-thick layer of carbon atoms arranged in a two-dimensional hexagonal lattice. The theoretical study of some of its properties goes back to the 1940's, however only in 2004 it has been experimentally realized. The experimental realization of graphene has spurred an enormous amount of interest and activity in the physics community due to the graphene unique properties and its possible applications in electronic devices.  One of the most interesting aspects of graphene is that the low energy electronic excitations are described by a massless Dirac fermion model. Electrons in graphene behave as ultra-relativistic electrons described by two-dimensional Quantum Electro Dynamics (QED), albeit with a much lower (1/300 th) speed of light and bigger ~ 1, (and tunable), fine structure constant.  After presenting the main properties of graphene I will discuss its unusual transport properties that arise from its gapless Dirac spectrum. Close to the Dirac point, where the average carrier density is zero, the physics of graphene is dominated by the density fluctuations.  I will present a microscopic theory that is able to characterize quantitatively these fluctuations. Finally I will discuss a transport theory for graphene that properly takes into account the strong density fluctuations close to the Dirac point and is able to answer semi-quantitatively some of the most puzzling questions that have been posed by transport experiments on graphene since its realization in 2004.