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CCAT-prime will be a 6-meter aperture telescope operating from sub-mm to mm wavelengths, located at 5600 meters elevation on Cerro Chajnantor in the Atacama Desert in Chile. Its novel crossed-Dragone optical design will deliver a high throughput, wide field of view capable of illuminating much larger arrays of sub-mm and mm detectors than can existing telescopes. We present an overview of the motivation and design of Prime-Cam, a first-light instrument for CCAT-prime. Prime-Cam will house seven instrument modules in a 1.8 meter diameter cryostat, cooled by a dilution refrigerator. The optical elements will consist of silicon lenses, and the instrument modules can be individually optimized for particular science goals. The current design enables both broadband, dual-polarization measurements and narrow-band, Fabry-Perot spectroscopic imaging using multichroic transition-edge sensor (TES) bolometers operating between 190 and 450 GHz. It also includes broadband kinetic induction detectors (KIDs) operating at 860 GHz. This wide range of frequencies will allow excellent characterization and removal of galactic foregrounds, which will enable precision measurements of the sub-mm and mm sky. Prime-Cam will be used to constrain cosmology via the Sunyaev-Zeldovich effects, map the intensity of [CII] 158 mu m emission from the Epoch of Reionization, measure Cosmic Microwave Background polarization and foregrounds, and characterize the star formation history over a wide range of redshifts. More information about CCAT-prime can be found at www.ccatobservatory.org.