Graphene, a two-dimensional material consisting of carbon atoms arranged in a honeycomb lattice, has become famous for the evidence that its electronic structure approximately corresponds to the one of massless Dirac fermions. However, in order to correctly describe graphene, the spin, which plays an essential role in the physics of Dirac fermions, has to be replaced by the so-called pseudospin, an intrinsic property of the honeycomb lattice which is not related to the electrons' real spin.If, now, the real spin is considered, too, the effective Hamiltonian has to be extended by terms which have no equivalents in the original Dirac Hamiltonian. While charge transport properties can be predicted from Dirac physics very realiably, the extended Hamiltonian leads to new phenomena in the context of spin transport.