Practically all known planet hosts will evolve into white dwarfs, and large parts of their planetary systems will survive this transition - the same is true for the solar system beyond the orbit of Mars. Spectroscopy of white dwarfs accreting planetary debris provides the most accurate insight into the bulk composition of exo-planets. Ground-based spectroscopic surveys of ~260, 000 white dwarfs detected with Gaia will identify >1000 evolved planetary systems, and high-throughput high-resolution space-based ultraviolet spectroscopy is essential to measure in detail their abundances. So far, evidence for two planetesimals orbiting closely around white dwarfs has been obtained, and their study provides important constraints on the composition and internal structure of these bodies. Major photometric and spectroscopic efforts will be necessary to assemble a sample of such close-in planetesimals that is sufficiently large to establish their properties as a population, and to deduce the architectures of the outer planetary systems from where they originated. Mid-infrared spectroscopy of the dusty disks will provide detailed mineralogical information of the debris, which, in combination with the elemental abundances measured from the white dwarf spectroscopy, will enable detailed physical modelling of the chemical, thermodynamic, and physical history of the accreted material. Flexible multi-epoch infrared observations are essential to determine the physical nature, and origin of the variability observed in many of the dusty disks. Finally, the direct detection of the outer reservoirs feeding material to the white dwarfs will require sensitive mid- and far-infrared capabilities.