Graphene - a monolayer of carbon atoms densely packed in a honeycomb lattice - was discovered by the applicants' group in 2004 and, since then, has established itself as one of the most remarkable materials available to condensed matter scientists today. It is only one atom thick but stable under ambient conditions and exhibits extraordinarily high crystal and electronic quality. The most important graphene physics originates from its very unusual electronic properties: In other conductors charge carriers are described quantum mechanically as electron waves obeying the Schrdinger equation (the wave equation of quantum physics) but in graphene electrons move according to the laws of relativistic quantum physics - the Dirac equation. Graphene has now become a real gold mine for searching for new fundamental phenomena, and it also offers numerous applications, ranging from smart materials to future electronics. Theory predicts a whole spectrum of magnetic phenomena in graphene, including several mechanisms for intrinsic ferromagnetism and spin-ordering effects that arise due to its low-dimensionality and Dirac-like spectrum. However, none of these effects has so far been explored experimentally. If confirmed, the existence of spin ordering in graphene will have important implications not only for understanding of this remarkable material but also for its various applications and the field of spintronics in general.We are also hoping that our experiments on graphene will help resolve the controversies surrounding recent findings of magnetism in the so-called unconventional magnetic materials (bulk graphite, fullerenes, some oxides and hexaborides). In these materials ferromagnetism has been detected despite the absence of any magnetic-moment-carrying atoms (usually ferromagnetic ordering requires the presence of atoms with partially filled d- or f- shells that have non-zero total spin) but the findings remain highly controversial and there are many uncertainties related to the presence of impurities and defects. Graphene, on the other hand, is an ultimately simple and clean experimental system (with no crystal defects and impurities) where the density of charge carriers can be controlled by gate voltage. Therefore it should allow unambiguous answers to questions related to magnetism in other graphitic materials, in which the inevitable presence of impurities and imperfections can obscure vital evidence or lead to artefacts.We have a unique combination of skills and experience to make this project a success. Indeed, the very small magnetic moments associated with the discussed phenomena, together with submicron-scale magnetic field gradients, make it very difficult to probe the signatures of magnetism by traditional methods but they can be readily observed using the high-resolution Bitter decoration technique available to the applicants. This technique has sufficient field sensitivity and wide temperature range and is known to provide unique - yet easy-to-interpret - information. We have established and been using the technique successfully to study submicron-scale magnetisation patterns in superconductors and some magnetic materials. The applicants' group also remains the world leader in studies of the physics and technology of graphene. We believe that the combination of the unique method and expertise, a new approach to the problem of magnetism without magnetic ions , and a new experimental way of studying the exceptional experimental system should ensure exceptional research outcome.