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Graphene is a material of considerable current interest owing to its linear band structure and excitations that behave as massless Dirac fermions. In this talk, I will focus on the physics of a vacancy in graphene and show that it forms a magnetic center and, quite interestingly, it is also a Jahn-Teller center due to the coupling between the vacancy electronic states and the local lattice modes. However, the energetics are such that there is only a small potential barrier between the Jahn-Teller minima, leading to the quantum mechanical tunneling of the nuclei between the three minima, resulting in the dynamical Jahn-Teller effect. The Berry phase introduced into the electronic motion leads to observable effects such as the symmetry of the nuclear ground state, which can be measured by EPR and two-photon scattering experiments. Signatures of the vacancy states seen in recent experiments will be discussed. We will also present results for interaction between two magnetic centers, the so-called RKKY interaction in solids. In 3D ordinary solids with quadratic band dispersion, the RKKY interaction oscillates between ferromagnetic and antiferromagnetic as a function of distance. Remarkably for graphene, it is always ferromagnetic or antiferromagnetic, depending on the sublattice locations of the two moments. The linear band structure also produces new features such as an interference term from the two Dirac cones in the Brillouin zone.