Since the first unambiguous detection of a planet around a Sun-like star, theinterest in the new and exciting field of exoplanets has grown immensely. Newand exciting developments are seen at a pace unparalleled for most subfieldsof astronomy. In this thesis, I describe the two most successful techniques forexoplanet detection and characterisation – transits and radial velocities – andthe challenges commonly encountered in extracting the planets from the data.Transit photometry allows us to measure the planet radius, while radialvelocity measurements give us the planet’s minimum mass. These methods’true strength, however, manifests in their combination as it allows us to estimatethe true mass, which, together with the radius, gives us the planet’s bulkdensity. This is a powerful quantity, which allows us to construct models andmake predictions about the structure and composition of a planet’s interior,as well as its atmosphere. Zeroing in on the latter two is currently one of thebiggest challenges for exoplanet characterisation.I describe the process of detecting a planet in a stellar light curve, andhow transits and radial velocities are modelled together in order to determinethe planet parameters. This is then followed by the ideal theoreticalapproach, which can be used to study a system in practice. However, the currentchallenges in exoplanet characterisation surpass the ideal case, leading usto explore more complex models. I then discuss the biggest nemesis to planetdiscovery, particularly in radial velocity timeseries – stellar activity, and theproblem of its often stochastic manifestation. A special focus is given to onemethod for its mitigation – modelling the radial velocities alongside activityindicators. This is the core concept of multi-dimensional Gaussian processregression, particularly with the quasi-periodic covariance function, which isused in a large part of this work.Finally, the last part if the thesis shows that while the ideal planet case cansometimes be applicable for quiet stars, as is the case of the TOI-2196 system,extending to non-parametric models, such as Gaussian processes, can help usto detect planets in complicated datasets, as demonstrated by the cases of theTOI-1260, TOI-733, TOI-776 and TOI-1416 systems
