Atomically thin materials, which were discovered in 2004 with the isolation of graphene, represent a unique new building block for mechanical systems. Here, we explore the possibilities for a new class of mechanical devices based on these materials, with a focus on graphene-based resonant nanoelectromechanical systems. As a result of the recent progress in fabricating large-area graphene sheets, graphenebased mechanical devices have become vastly easier to manufacture and now show even greater promise for a range of applications, such as signal processing, sensing, and mechanical systems in the quantum regime. We discuss recent advances in fabrication and measurement techniques that make graphene resonators a viable technology, and present what is known about the performance of graphene mechanical systems. We demonstrate in the area of performance that room temperature graphene quality factor can be enhanced an order of magnitude by using a fully-clamped circular geometry with large diameter. We also show that the quality factor of a graphene resonator depends on its tension, and we suggest ways of enhancing the tension in graphene resonators. Finally, we simultaneously utilize graphene's electrical, mechanical, and optical properties in a novel application of graphene: photothermal graphene optomechanics. As a demonstration of the utility of this effect, we show that a continuous wave laser can be used to cool a graphene vibrational mode or to power a graphene-based tunable-frequency oscil- lator. By virtue of graphene's high thermal conductivity and optical absorption, photothermal optomechanics is efficient in graphene and could ultimately enable laser cooling of graphene mechanical systems to the quantum ground state or applications such as photonic signal processing.