The reflection of the X-rays emitted from a corona of energetic particles surrounding an accreting black hole from the accretion disc is investigated in the context of probing the structure of the central regions as well as the physical processes that power some of the brightest objects seen in the Universe. A method is devised to measure the emissivity profile of the accretion disc, that is the reflected flux as a function of radius in the disc. This method exploits the variation in the Doppler and gravitational redshift of emission from different radii in the disc to fit the observed reflection spectrum as the sum of contributions from successive radii and is applied to X-ray spectra of the narrow line Seyfert 1 galaxies 1H 0707-495, IRAS 13224-3809 and MCG-6-30-15 as well as the Galactic X-ray binary, Cygnus X-1. This illumination pattern of the accretion disc is a sensitive probe of the geometry of the corona that is illuminating the disc. A formalism is developed in which systematic ray tracing simulations can be run between X-ray emitting coronae and the accretion disc for a range of source geometries and other physical parameters, allowing observable data products to be simulated that can be directly compared to data from astrophysical black holes, in order to determine how these parameters affect the observed data, allowing them to be constrained observationally. The measured emissivity profiles are found to be in agreement with those expected theoretically and it is also discovered that the measured emissivity profile can be used to determine the radial extent of the X-ray emitting corona above the accretion disc. The X-ray emitting coronae are located and their radial extents constrained in 1H 0707-495, IRAS 13224-3809 and MCG-6-30-15, while the insight gained into accretion disc emissivity profiles from ray tracing simulations allows the low flux state that 1H 0707-495 was seen to drop in to in January 2011 to be explained in terms of a collapse of the X-ray emitting corona to a confined region around the central black hole. The rapid variability of the X-ray emission from accreting black holes is exploited in the use of reverberation time lags, where variability in the continuum is seen to lead that in its reflection from the accretion disc, to measure the distances between the X-ray emitting corona and the reflector. Ray tracing calculations are developed to simulate lag spectra that can be measured in X-ray observations to provide a means of constraining the extent and geometry of the corona, complimentary to the use of the emissivity profiles. Combining these methods, the X-ray emitting coronae are constrained to extend radially outward a few tens of gravitational radii over the accretion disc, while extending vertically a few gravitational radii above the plane of the disc. Furthermore, it is demonstrated how measured lag spectra can be used to understand the propagation of luminosity fluctuations through the extent of the corona and techniques are developed for analysing energy-resolved variability analysis that will be possible with future generations of X-ray telescopes. Finally, these methods, along with theoretical insight gained form ray tracing simulations, are applied to X-ray spectra extracted from 1H 0707-495 during periods of low and high flux during the observations. Evidence is found for the expansion of the corona along with a drop in the average energy density as the X-ray luminosity increases followed by its contraction as the luminosity decreases on timescales of hours.