Very recently, a new class of sub-luminous accreting neutron stars and black holes has been identified. I propose to use these objects to probe the extreme physical processes which are associated with such compact stars. Just as with their better known brighter cousins, studying them when they are actively accreting and when they are in their quiescent states will give us clues about the behavior of ultra-dense matter in neutron stars and the way neutron-star magnetic fields decay due to the accretion of matter. However, given that these new systems behave differently, I expect to derive from their study a novel perspective which will gain in value even further when contrasted with our current knowledge. I further believe their study will allow me to significantly strengthen the observational proof for the presence of event horizons in black holes. The uncommon nature of these systems suggests that they are very unusual outcomes of binary evolution, and I expect this will also provide us with a different set of clues than we have had until now about the formation of binaries which harbor compact stars. These objects have only recently been discovered, both because we did not have the sensitivity to see them, and because we did not know how to optimize our searches to find them. Current instruments finally have reached the necessary sensitivity. I propose new approaches to find and study these sub-luminous systems using these X-ray and radio instruments in combination with multi-wavelength studies. I expect to find these systems in greater numbers than before, allowing systematic studies of their properties which in turn will provide the ingredients needed to investigate the fundamental physics associated with neutron stars and black holes and serve as input for my proposed theoretical study into binary evolution.