When stars die, they form white dwarfs, neutron stars or black holes. These are key objects in astrophysics as their extreme gravities, densities and pressures allow us to test our theories of fundamental physics at the limits of their predictive powers. One of the best ways of studying white dwarfs, neutron stars and black holes is via their variations in brightness. Unfortunately, due to the small physical sizes of these objects, they are extremely faint and their variability occurs on timescales of milliseconds to seconds, too fast to be recorded by the current generation of astronomical instruments on the world's largest telescopes.What is required is a new type of astronomical instrument with the capability to take high-speed exposures with no noise from either the detector or the atmosphere. Such an instrument would enable us to answer some of the most important questions in astrophysics, such as: What are the progenitors of type Ia supernovae? What is the equation of state of the degenerate matter found in white dwarfs and neutron stars? What is the nature of the flow of matter close to the event horizon of black holes? What gravitational wave signals are likely to be detected by the next generation of space and ground-based detectors?My aim is to answer the above questions by building and exploiting a new astronomical instrument: HiPERCAM. HiPERCAM will be by far the best high-speed camera in the world, giving an order of magnitude improvement in performance over what has come before. It will be mounted on the world's best telescopes, including the 8.2-m VLT in Chile and the 10.4-m GTC on La Palma. This revolutionary new camera will incorporate a novel scintillation-correction mechanism and the latest in low-noise, high-speed detector technology. Although challenging, my track record with previous related instruments (e.g. ULTRACAM) and my preceding technology demonstration projects shows that my proposed research programme is feasible.