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This paper introduces the novel field of nanoelectromechanical quantum circuits and systems. The field derives from exploiting progress in techniques for fabricating, down to nanometer-length scales, freestanding device structures that incorporate mechanical motion and that may be designed to perform a variety of functions, such as optical, electrical, and, in particular, mechanical and mixed domain. The ability to create these nanomechanical structures, in turn, brings within our reach a tremendous possibility for both creating superior implementations of conventional circuits and systems, as well as entirely new ones. Since novel quantum mechanical effects, for instance, quantized heat flow, manifestation of charge discreteness, and the quantum electrodynamical Casimir effect, become operative in this regime, exciting new paradigms for circuit modeling and design must be invoked in order to fully exploit the potential of this technology in sensing, computation, and signal processing applications.