Spectroscopy of transiting exoplanets has revealed a wealth of information about their atmospheric compositions and thermal structures. In particular, studies of highly irradiated exoplanets at temperatures much higher than those found in our solar system have provided detailed information on planetary chemistry and physics because of the high level of precision which can be obtained from such observations. In this dissertation I use a variety of observation and modeling techniques to study the atmospheres of highly irradiated transiting exoplanets and learn about their physics and chemistry. In the first part of my thesis I present a detailed study of ultra-hot Jupiters, planets with equilibrium temperatures above 2000 K. I present Hubble Space Telescope (HST) secondary eclipse observations of the ultra-hot Jupiter HAT-P-7b, which shows a featureless emission spectrum. This spectrum, in combination with similar observations of other ultra-hot Jupiters, led to the realization that the ultra-hot Jupiters are a distinct class of planets with spectra heavily impacted by molecular dissociation. I next present a Spitzer Space Telescope phase curve of KELT-9b, the hottest known exoplanet, which shows the impacts of extreme molecular dissociation at the highest temperatures. KELT-9b is so hot that molecular hydrogen dissociates and significantly impacts the heat transport throughout its atmosphere. Finally, I expand these initial investigations to a population study of all hot Jupiter emission spectra observed with HST to investigate the impact of molecular dissociation across a wider range of temperatures. In the second part of my thesis I investigate the physics and chemistry of the highly irradiated exo-Neptune HAT-P-11b. I present a discovery of helium in the HST transmission spectrum of HAT-P-11b. This discovery was only the second time helium was detected using HST, and it allows us to probe atmospheric escape at the uppermost levels of the atmosphere and study the history of atmospheric loss on the planet. In the coming years we will have the opportunity to study exoplanet atmospheres in even more detail using the James Webb Space Telescope (JWST). In the final section of my thesis I present modeling tools I have developed which will be key to interpreting future JWST exoplanet observations. I first present a data reduction and analysis pipeline that can be used to interpret JWST eclipse mapping observations of hot Jupiters, which will give us the ability to resolve the daysides of these planets in latitude, longitude, and altitude. I next turn to smaller planets and present a model of inferred albedos for hot, rocky planets which can be used to determine whether such planets have atmospheres, a key prerequisite for the presence of life.