Since the first discovery of a transiting planet in 2000, transmission spectroscopy has proved essential for characterising the rapidly increasing number of known extrasolar planets. When a planet is in a favourable alignment, it periodically passes (transits) in front of its host star, during which time it blocks a fraction of the stellar light. During a transit, the starlight passes through the planetary atmosphere, causing the signatures of atoms or molecules present in that atmosphere to imprint themselves on the stellar spectrum, allowing direct observation of a planet's atmospheric composition. At the start of this thesis, only two planets (HD 189733b and HD 209458b) had been studied in any detail, mainly from space. The two planets showed surprisingly different qualities for two objects with only a small temperature difference between them, and motivated both wider and more detailed studies of the exoplanet population. Since the start of my PhD, the amount of exoplanet knowledge has grown rapidly, with observations from the ground becoming important, and with studies branching out towards new planets. There are several contributions made by this thesis to the field. Chapter 3 details the detection of the resolved sodium D doublet in the atmosphere of HD 189733b, a planet with a featureless broad-band transmission spectrum dominated by Rayleigh scattering. The results confirmed the presence of sodium absorption as well as resolving the feature for the first time, and placing constraints on relative abundances. Furthermore, in Chapter 4, I outline a method based on earlier work which allows observers to retrieve atmospheric temperature information from resolved spectral features. This method is applied to the observations of HD 189733b, showing that the planet has a hot thermosphere similar to HD 209458b. The models are then also used in later chapters. I then present the first results from a ground-based optical long-slit spectroscopic survey in Chapter 5, and the first results from a space-based optical-near-IR spectroscopic survey in Chapter 6. From the ground, I detect absorption from sodium in the atmosphere of XO-2b, making this the first planet with sodium and potassium detected in its atmosphere. I also find that the Na I D feature lacks broad line wings, suggesting haze or cloud cover. From space, I observed the transmission spectrum of WASP-19b, finding solar abundance water features and a likely lack of predicted TiO features. WASP-19b is the first planet to have confirmed water features at solar-abundance level. In Chapter 7 I conclude and discuss future work, including a project aimed at understanding why WASP-19b lacks TiO features, and projects which move beyond the hot Jupiter class.

The brightness and variability of the atmosphere in the thermal infrared poses obstacles to precision photometry measurements. The need to remove atmospheric effects calls for the use of a comparison star, but it is usually impossible to fit both science and comparison targets on current long-wavelength (> 2 mu m) detectors. We present a new pointing mode at the Large Binocular Telescope, which has twin 8.4-m primary mirrors that can be pointed up to similar to 2 arcminutes apart and allow the placement of both targets on a small-field infrared detector. We present an observation of the primary transit of an exoplanet in front of its host star, and use it to provide preliminary constraints on the attainable photometric precision.