In this thesis we study the mass-radius and pressure-radius relations of white dwarfs and neutron stars. We derive the pressure and energy density in a non-interacting Fermi gas at $T=0$, and construct several different equations of state. A comparison is made between different numerical algorithms, and a fourth order Runge-Kutta method is found to be the most suitable option. The Newtonian equations of stellar structure are derived, and then solved numerically to find a the mass-radius relation and maximum mass for white dwarfs. Good agreement with observational data is established. We derive the Tollman-Oppenheimer-Volkoff equation and solve them numerically to find the mass-radius relation and maximum mass for neutron stars. The maximum mass found appears to be significantly lower than expected. Finally the issue of stability and collapse is addressed.