Atmospheric spectroscopy of planets beyond our solar system can provide important insights into their atmospheric processes, formation histories and even the presence of life. The James Webb Space Telescope (JWST) has already led to the first of likely many breakthroughs with novel chemical detections. These expected advances in exoplanetary science in the JWST era necessitate atmospheric retrieval techniques to evolve to take full advantage of JWST observations. This thesis contributes towards that objective, considering what atmospheric inferences are possible through transmission spectroscopy with JWST and the advances in atmospheric retrieval techniques necessary to achieve them.

I first present what atmospheric abundance constraints are achievable with JWST observations of cloudy, temperate sub-Neptunes. Considering two exoplanets as case studies and several instrument configurations, I find that JWST observations of suitable exoplanets over a broad wavelength range can yield precise abundance constraints for prominent molecules, even in the presence of high altitude clouds.

I then consider atmospheric retrievals with the first JWST observations of the hot Saturn WASP-39 b, spanning a previously unexplored spectral region. I implement a physically-based Mie scattering calculation to model the spectral contributions of clouds. Using this new aerosol model, I constrain the atmospheric elemental abundances of O, C and S, finding them to be largely consistent with the inferred C abundance of Saturn.

I subsequently present VIRA, a new atmospheric retrieval framework designed to make the most of JWST observations. VIRA implements several complementary models for atmospheric composition, aerosols and temperature structure, as well as rigorously accounting for correlated observations. I use VIRA to analyse JWST observations of WASP-39 b, confirming prior results while also finding spectral contributions from ZnS aerosols and robustly inferring CH₄ depletion.

I also present work carried out as a contribution to recent studies. This includes analyses of observations from space and ground for a variety of gas giant planets, assessing their compositions and impact of clouds and stellar heterogeneities. In doing so, I identify a number of themes that are characteristic of the last decade of transmission spectroscopy, which are set to persist in the JWST era.

Lastly, I present work examining the observability of several chemical species in the atmospheres of Hycean planets. This work demonstrates that by taking advantage of the comparatively observable atmospheres of Hycean planets, a number of candidate biomarker species are readily detectable by JWST.

This thesis highlights the wealth of information that can be encoded in precise observations of exoplanetary transmission spectra. At the same time, it demonstrates the pivotal importance of sophisticated and robust atmospheric retrievals in understanding the atmospheres, interiors and formation histories of exoplanets and the search for life beyond Earth.