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The last two decades have witnessed a tremendous growth in the development and understanding of carbon-based nanostructures. The impact of this research has led to a number of fundamental discoveries that have played a central role in the understanding of many aspects of materials physics and their applications. Much of this progress has been enabled by the development of new techniques to prepare, modify, and assemble low-dimensional materials into devices. The field has also benefited greatly from much progress in theoretical and computational modeling, as well as from advances in characterization techniques developed to probe and manipulate single atomic layers, nanoribbons, and nanotubes. Some of the most fundamental physical properties of carbon-based nanostructures are reviewed and their role as model systems for solid-state physics in one and two dimensions is highlighted …