The study of planets orbiting other stars, exoplanets, has entered the era of characterisation. When an exoplanet passes in front, or transits, its parent star, absorption features are imprinted into starlight passing through the planetary atmosphere. By analysing the resultant transmission spectrum, one can reveal the chemical composition of these distant worlds. Already, this technique has yielded detections of multiple chemical species in exoplanetary atmospheres. However, measuring the abundances of these molecules and atoms, as required to infer exoplanet formation mechanisms, has been challenged by the prospect of atmospheric clouds.

In this thesis, I introduce a new approach to retrieve properties of exoplanetary atmospheres. I demonstrate that a generalisation of common 1D exoplanet atmosphere retrieval models to include 2D properties can break cloud-chemistry degeneracies. This algorithm, implemented by a new atmospheric retrieval code, POSEIDON, enables precise constraints on the chemistry, cloud properties, and temperature structure of exoplanetary atmospheres. By applying POSEIDON to observed transmission spectra of giant exoplanets, new insights into exoplanet atmospheres have been obtained. I present evidence of inhomogeneous clouds and disequilibrium nitrogen chemistry (NH$3$ and HCN) in hot Jupiter atmospheres, the first detection of titanium oxide (TiO) in an exoplanetary atmosphere, and evidence of metal hydrides, possibly from a recent impact, in the atmosphere of a Neptune-mass exoplanet. Finally, looking to the future, I examine the ability of the upcoming James Webb Space Telescope to probe the atmosphere of a Neptune-mass exoplanet in unprecedented detail.

These results offer extraordinary promise for the retrieval of atmospheric properties from exoplanet spectra. Planets possessing cloudy skies can still be precisely characterised, expanding the potential for new discoveries in the years to come.