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We propose to utilize multiple-graphene-layer structures with lateral p-i-n junctions for terahertz and infrared (IR) photodetection and substantiate the operation of photodetectors based on these structures. Using the developed device model, we calculate the detector dc responsivity and detectivity as functions of the number of graphene layers and geometrical parameters and show that the dc responsivity and detectivity can be fairly large, particularly, at the lower end of the terahertz range at room temperatures. Due to relatively high quantum efficiency and low thermogeneration rate, the photodetectors under consideration can substantially surpass other terahertz and IR detectors. Calculations of the detector responsivity as a function of modulation frequency of THz and IR radiation demonstrate that the proposed photodetectors are very fast and can operate at the modulation frequency of several tens of gigahertz.