Exoplanet atmosphere characterization has the potential to reveal the origins, nature, and even habitability of distant worlds. This thesis represents a step towards realizing that potential for a diverse group of four extrasolar planets. Here, I present the results of intensive observational campaigns with the Hubble and Spitzer Space Telescopes to study the atmospheres of the super-Earth GJ 1214b and the hot Jupiters WASP-43b, WASP-12b, and WASP-103b. I measured an unprecedentedly precise near-infrared transmission spectrum for GJ 1214b that definitively reveals the presence of clouds in the planet’s atmosphere. For WASP-43b and WASP-12b, I also measured very precise spectra that exhibit water features at high confidence (>7 sigma). The retrieved water abundance for WASP-43b extends the well-known Solar System trend of decreasing atmospheric metallicity with increasing planet mass. The detection of water for WASP-12b marks the first spectroscopic identification of a molecule in the planet's atmosphere and implies that it has solar composition, ruling out carbon-to-oxygen ratios greater than unity. For WASP-103b, I present preliminary results from the new technique of phase-resolved spectroscopy to determine the planet's temperature structure, dynamics, and energy budget. In addition to these observations, I also describe the BATMAN code, an open-source Python package for fast and flexible modeling of transit light curves. Taken together, these results provide a foundation for comparative planetology beyond the Solar System and the investigation of Earth-like, potentially habitable planets with future observing facilities.