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In this thesis, graphene (i.e. monolayer carbon film) and carbon clusters supported on a transition metal surface are systematically studied by local probe techniques, with respect to their structures, electronic properties and formation mechanisms. The main tools used are low-temperature scanning tunnelling microscopy and spectroscopy (STM and STS), which are introduced in Chapter 2. The mechanism of the resonance tunnelling at electron energies higher than the work function of the surface is discussed in detail, and a qualitative explanation of the Gundlach oscillations in the corresponding spectroscopy is presented. Epitaxial graphene synthesised on the Rh(111) surface by ethylene dehydrogenation is investigated by STM in Chapter 4. Such carbon film exhibits a hexagonal Moiré pattern due to a lattice mismatch between graphene and the rhodium substrate. The periodicity and local registries of the graphene/Rh(111) superstructure are carefully analysed. Based on a thorough discussion about the “commensurate vs. incommensurate” nature of the Moiré pattern in surface science field, the graphene/Rh(111) system is identified to have a non-simple-commensurate superstructure. The surface electronic properties and geometric buckling of graphene/Rh(111) are investigated by resonance tunnelling spectroscopy (RTS) and density functional theory (DFT) calculations in Chapter 5. Spectroscopy measurements reveal a modulation of the electronic surface potential (or work function Φ) across the supercell of epitaxial graphene. Based on the microscopy/spectroscopy data and the extended DFT calculations, we examined the electronic coupling of the various local C-Rh registries, and identified both experimentally and theoretically the local atomic configurations of maximum and minimum chemical bonding between graphene and the rhodium substrate. We studied in Chapter 6 the growth mechanism of graphene on Rh(111) at elevated temperatures. This part starts by investigating the dehydrogenation of ethylene into ethylidyne. When the dehydrogenation process is complete, monodispersed carbon species, identified as 7C6, are found to dominate the cluster population on the rhodium terraces. A significant coalescence of the 7C6 clusters into graphene islands occurs at temperatures higher than 873 K. The structural and electronic properties of the 7C6 carbon clusters are examined by high-resolution STM and STS, and compared with coronene molecules, i.e. the hydrogenated analogues of 7C6. DFT calculations are further used to explain the stability of 7C6 supported on the Rh(111) surface, and also the structural characteristics of such magic-sized carbon clusters.