We review observations of a representative set of extrasolar planets that transit their stars, concentrating on those discovered and characterized by the XO Project. Spectra of these planets in transit and in eclipse have made significant contributions to our understanding of hot gas giant exoplanets, including 1) evidence for planet-planet scattering to transfer the planets from where they are formed to where we observe them, 2) hot stratospheres of these exoplanets, and two possible mechanisms to maintain them, and 3) water vapor detected in the near-IR spectrum of the exoplanet XO-1b in transit. For the latter case, we compare near-IR spectra obtained with two HST instruments: NICMOS and WFC3 with its new spatial scanning technique. We then present the spectrum of the super-Earth exoplanet GJ 1214b from the visible to the infrared, and focus on the definitive results obtained with HST WFC3 that show a featureless near-IR spectrum, indicative of either a large mean molecular weight in the planet's atmosphere, or obscuring haze (Berta et al. 2012). We identify similar observations that are being made with HST now, and will be made with JWST, and other telescopes in the future. We conclude by summarizing the Transiting Exoplanet Survey Satellite, TESS, which will discover the nearest, transiting rocky exoplanets, those most interesting and most suitable for follow-up characterization of the sort we have presented.