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From wispy gas giants on the verge of disruption to tiny rocky bodies already falling apart, short-period exoplanets pose a severe puzzle to theories of planet formation and orbital evolution. By far most of the planets known beyond the solar system orbit their stars in much tighter orbits than the most close-in planet in the solar system, Mercury. Short-period planets experienced dynamical and evolutions histories distinct from their farther-out cousins, and so it's not clear they are representative of all planets. These exoplanets typically have radii between about 1 and 4 Earth radii, whereas the solar system does not contain any planet in this radius range. And while the most massive planets in the solar system occupy the icy regions beyond about 5 AU from the sun, about 1% of sun-like stars have a Jupiter-mass planet near 0.05 AU, with just a few days of an orbital period. How did these short-period planets get there? Did they form in-situ, or did they migrate towards their contemporary orbits? If they migrated, what prevented them from falling into their stars? Vice versa, could some of the remaining 99% of stars without such a hot Jupiter show evidence of their past consumption of a close-in, massive planet? The proximity between short-period planets and their host stars naturally facilitates observational studies, and so short-period planets dominate our observational constraints on planetary composition, internal structure, meteorology, and more. This white paper discusses the unique advantages of short-period planets for the theoretical and observational investigations of exoplanets in general and of their host stars.