Compared to the Earth, the exoplanet Kepler-78b has a similar size (1.2 R⊕) and an orbital period a thousand times shorter (8.5 h). It is currently the smallest planet for which the mass, radius, and dayside brightness have all been measured. Kepler-78b is an exemplar of the ultra-short-period (USP) planets, a category defined by the simple criterion Porb < 1 day. We describe our Fourier-based search of the Kepler data that led to the discovery of Kepler-78b, and review what has since been learned about the population of USP planets. They are about as common as hot Jupiters, and they are almost always smaller than 2 R⊕. They are often members of compact multi-planet systems, although they tend to have relatively large period ratios and mutual inclinations. They might be the exposed rocky cores of "gas dwarfs," the planets between 2-4 R⊕ in size that are commonly found in somewhat wider orbits.