In this thesis I have studied how discs around young stars evolve and disperse. In particular, I build models which combine viscous evolution with photoevaporation, as previous work suggests that photoevaporation can reproduce the observed disc evolution and dispersal time-scales. The main question this thesis attempts to address is: Can photoevaporation provide a dominant dispersal mechanism for the observed population of young stars? Photoevaporation arises from the heating that high energy (UV and X-ray) photons provide to the surface layers of a disc. Before I started this work, only photoevaporation from a pure EUV radiation field was described within a hydrodynamic framework. Therefore, I start by building a hydrodynamic solution to the pure X-ray photoevaporation problem, and then extend this solution to the entire high energy spectrum. This hydrodynamic model leads me to conclude that it is the X-ray radiation field that sets the mass-loss rates. These mass-loss rates scale linearly with X-ray luminosity, are independent of the underlying disc structure and explicitly independent of stellar mass. I build a radiation-hydrodynamic algorithm, based on previous work, to describe the process of X-ray heating in discs. I then use this algorithm to span the full range of observed parameter space, to fully solve the X-ray photoevaporation problem. I further extend the algorithm to roughly approximate the heating an FUV radiation field would have on the photoevaporative flow, as well as separately testing the effect an EUV radiation field will have. These numerical tests are in agreement with the hydrodynamic model derived. Specifically, it is the X-rays that are driving the photoevaporative flow from the inner disc. Armed with an accurate description of the photoevaporative mass-loss rates from young stars, I consider the evolution of a population of disc-bearing, young (