The last evolutionary stage of about 97 per cent of the stars in the Milky Way is the white dwarf phase: a remnant stellar core sustained by electron degeneracy pressure with a typical mass of 0.6 solar masses and a radius such as the Earth's. The study of white dwarfs is not only genuinely interesting due to them being the remnants of the majority of stars, but also because they are important to different astrophysical fields, such as the theory of stellar evolution as tracers of Galaxy star formation or the theory of stellar interactions (a significant fraction of white dwarfs are found in binary systems). Despite these reasons being more than enough to grant these remnants a prominent role in modern astrophysics, they also provide the opportunity to reveal the chemical composition of exoplanets, currently unfeasible in any other way. In fact, we now know that about 50 per cent of white dwarfs may show metal pollution arising from the accretion of disrupted planetesimals. Therefore, the chemical analysis of their photospheres can lead to the bulk composition of the parent bodies in a direct way.

In the last few years, increasing evidences of evolved planetary systems around white dwarfs have been found: metal pollution of their otherwise pure hydrogen and/or helium photospheres as a result of the accretion of planetesimals, an infrared excess caused by dusty circumstellar debris discs, evolving, asymmetric transiting signals arising from the ongoing disruption of planetary bodies or emission lines originated in circumstellar gas discs.

The study of evolved planetary systems can render powerful insights into the chemistry of exoplanets, the behaviour and lifetime of circumstellar debris and gaseous discs and the unprecedented opportunity to explore disruption processes in real time.

This thesis focuses on the search and characterisation of evolved planetary systems around white dwarfs, which will serve to probe into the ultimate fate of our Solar System. In this framework, we present a theoretical introduction of exoplanets, white dwarfs and current status of the field in Chapter 1 and a summary of the methodology used throughout this thesis in Chapter 2.

Among the evolved planetary systems, a still scarce but outstanding subgroup is composed of the transiting white dwarfs, whose light curves show periodic dimmings as a result of planetary bodies passing in front of the star along our line of sight. So far, we know of just three such white dwarfs but they have the prospect to provide valuable knowledge into the hierarchy of evolved planetary systems. In Chapter 3, we present simultaneous spectrophotometry and photometry of the first transiting white dwarf detected, WD 1145+017. We determined its fundamental parameters by means of survey photometric magnitudes, confirmed that the transits in the optical are achromatic (i.e. grey) by analysing the deepest transit ever recorded, postulated an optically-thick dust cloud to account for the non-detection of small, micron-sized particles and detected a shallower circumstellar absorption line during this transit, that supports a spatial correlation between the dust and the gas discs. Finally, we developed an innovative tool to model the substructure seen at minimum transit flux, which we suggest is due to a train of overlapping dust clouds equally spaced in azimuth or a debris cloud with inhomogeneities.

We also investigate metal-polluted, helium-dominated white dwarfs, which are more difficult to characterise than their hydrogen counterparts, but display compelling singularities that are possibly linked to accreted water-rich planetesimals, of undeniable significance in the framework of habitable exoplanets. To achieve this, we have exploited state-of-the-art white dwarf model atmospheres to fit spectroscopic and photometric data of a number of these white dwarfs, with the ultimate goal of deriving the bulk composition of the parent bodies responsible for the photospheric pollution.

The analysis of GD 424, a heavily metal-polluted, helium-dominated white dwarf with large traces of hydrogen, is presented in Chapter 4. We determine its atmospheric parameters using a hybrid technique which fits spectroscopic and photometric data simultaneously. We also discuss the chemical composition of the likely engulfed planetesimal in terms of the time elapsed since accretion started or stopped (no obvious reason was found to assume the actual accretion state of the source). The results show that the pollution was most likely acquired in a recent or ongoing accretion event and that the bulk composition of the parent body resembles that of CI chondrites and the bulk Earth. We also compute the oxygen budget by assuming the same mineralogy as in the Solar System, finding no oxygen excess, and thus linking the large amount of hydrogen in the photosphere to earlier accretion of water-bearing planetesimals.

Finally, in Chapter 5, we present the study of 13 metal-polluted, helium-dominated white dwarfs. We use multiple spectroscopy and photometry data of each of these sources to get a realistic estimate of the systematic uncertainties of the photospheric parameters published in the literature. The use of different data sets produces mean differences of up to a thousand Kelvin, 0.27 dex and 0.23 dex for the effective temperature, surface gravity and hydrogen to helium relative abundance, respectively. In addition, we study the systematic uncertainties that the use of unrealistic chemical composition synthetic models may introduce. To this end, we fit our data with pure helium, helium+hydrogen and helium+hydrogen+metals synthetic spectra. We perform these tests for the spectroscopic and photometric techniques and find mean differences in the effective temperature of 400 K and of 0.04 dex for the surface gravity. These differences are, in both cases, significantly larger than the uncertainties published in independent studies, which points to a general overestimate of the  accuracy in the atmospheric parameters of white dwarfs.