The study of compact binaries invokes core astrophysical concepts ranging from stellar and sub-stellar atmospheres and interiors, stellar and binary evolution to physics of accretion. All of these systems are hosts to a compact object a white dwarf,neutron star or black hole - which produces a wide variety of exotic and energetic phenomena across the full electromagnetic spectrum. In this thesis, I will make use of multi-wavelength observations ranging from far-ultraviolet to near-infrared in order to investigate two main topics: a) the late evolution of cataclysmic variables, and b) the accreting state of transitional millisecond pulsars.

Firstly, I analyse the Very Large Telescope X-Shooter time-resolved spectroscopy of the short orbital period cataclysmic variable, SDSS J1433+1011,in Chapter 2. The wide wavelength coverage allowed me to perform a detailed characterisation of the system, as well as a direct mass measurement of the brown dwarf companion. I show that the donor in SDSS J1433+1011 successfully transitioned from the stellar to sub-stellar regime, as predicted by evolutionary models. Further light-curve modelling allowed me to show that allow albedo as well as a low heat circulation efficiency is present in the atmosphere of the sub-stellar donor. In Chapter 3, I analyse data from large synoptic surveys, such as SDSS and PTF, to search for the predicted population of dead cataclysmic variables. Following the non-detection of dead CVs, I was able to estimate the space density (p0 < 2 x 10^-5 pc^-3) of this hidden population via a Monte Carlo simulation of the Galactic CV population. In Chapter 4, I present Hubble Space Telescope ultraviolet observations of the transitional millisecond pulsar PSR J1023+0038, during its latest accretion state. In combination with optical and near-infrared data, I show that a standard accretion disc does not reach the magnetosphere of the neutron star. Instead,the overall spectrum is consistent with a truncated disc at ∼ 2:3 x 10⁹ cm away from the compact object. Furthermore, the ultraviolet data shares remarkable similarities with the only accreting white dwarf in a propeller regime, AE Aqr. Finally, I summarise my results in Chapter 5 and provide future lines of research in accreting compact binaries based on this work.