The characterisation of exoplanets has made rapid progress in recent years, with observations of bulk properties such as mass and radius combining with detailed atmospheric spectroscopy to provide unprecedented insight into the nature of these remote worlds. However, these high-quality observations also require sophisticated modelling and analysis tools in order to maximise the scientific output from the data. In this thesis I present a number of advances in atmospheric modelling and retrieval, as well as internal structure models, which have been used to investigate the properties of a wide range of planets, from hot Jupiters to temperate mini-Neptunes.

I conduct an assessment of the feasibility of supervised machine learning as a tool to carry out atmospheric retrievals of exoplanets. Retrieval methods commonly conduct Bayesian parameter estimation and statistical inference using sampling algorithms such as Markov Chain Monte Carlo or Nested Sampling. Recently several attempts have been made to use machine learning algorithms either to complement or replace fully Bayesian methods in order to improve computational efficiency. However, results from these algorithms sometimes disagree with contemporary Bayesian retrievals. To investigate this, I use the Random Forest supervised machine learning algorithm which has been applied previously for atmospheric retrieval. I extend the machine learning approach to develop a new algorithm, and demonstrate excellent agreement with a Bayesian retrieval of the transmission spectrum of the hot Jupiter HD~209458b. Despite this success, and achieving high computational efficiency, I still find that this machine learning approach is computationally prohibitive for high-dimensional parameter spaces that are routinely explored with Bayesian retrievals with modest computational resources. I discuss the trade offs and potential avenues for the future.

I present \textsc{Aura-3D}, a three-dimensional atmospheric retrieval framework for exoplanet transmission spectra. \textsc{Aura-3D} includes a forward model that enables rapid computation of transmission spectra in 3D geometry for a given atmospheric structure and can, therefore, be used for atmospheric retrievals as well as for computing spectra from General Circulation Models (GCMs). In order to efficiently explore the space of possible 3D temperature structures in retrievals, I develop a parametric 3D pressure-temperature profile which can accurately represent azimuthally-averaged temperature structures of a range of hot Jupiter GCMs. I apply this retrieval framework to simulated JWST observations of hot Jupiter transmission spectra, obtaining accurate estimates of the day-night temperature variation across the terminator as well as the abundances of chemical species. I demonstrate an example of a model hot Jupiter transmission spectrum for which a traditional 1D retrieval of JWST-quality data returns biased abundance estimates, whereas a retrieval including a day-night temperature gradient can accurately retrieve the true abundances. The forward model also has the capability to include inhomogeneous chemistry as well as variable clouds/hazes. This new retrieval framework opens the field to detailed multidimensional atmospheric characterisation using transmission spectra of exoplanets in the JWST era.

I also present a new internal structure model for super-Earths and mini-Neptunes that enables detailed characterisation of a planet's water component. I use my model to determine how the bulk properties and surface conditions of a water world affect its ocean depth, finding that oceans can be up to hundreds of times deeper than on Earth. I explore the region of mass--radius space in which planets with H-rich envelopes could host liquid H$2$O. Such envelopes could contribute significantly to the planet radius while retaining liquid water at the surface, highlighting the exciting potential for habitable conditions to be present on planets much larger than Earth.

I contribute to internal structure models of a number of sub-Neptunes whose atmospheres are set to be observed using JWST. Before such observations take place, it is vitally important to understand the interior structures of these planets, which strongly affects their possible atmospheric compositions. We use the bulk parameters and retrieved atmospheric properties to constrain the internal structure and thermodynamic conditions in the habitable-zone mini-Neptune K2-18b, for which I contribute the H$2$O EOS. The constraints on the interior allow multiple scenarios between rocky worlds with massive H/He envelopes and water worlds with thin envelopes. We constrain the mass fraction of the H/He envelope to be $\lesssim$6%; spanning $\lesssim$10$^{-5}$ for a predominantly water world to $\sim$6\%forapureironinterior.ThethermodynamicconditionsatthesurfaceoftheH$_2$O layer range from the supercritical to liquid phases, with a range of solutions allowing for habitable conditions. We also investigate the possible compositions of the pair of planets orbiting the star TOI-776. The bulk densities of TOI-776b and c allow for a wide range of possible interior and atmospheric compositions. However, the models indicate that both planets must have retained a significant atmosphere. Upcoming observations will revolutionise our understanding of these planets, helping to uncover the mysteries of the sub-Neptune population.

Finally, I discuss the latest developments in exoplanet observations, and consider how these advances may further our understanding of worlds beyond our own.