Abstract: The inverse stellar structure problem determines the equation of state of the stellar matter from a knowledge of the stars' observable macroscopic properties, like their masses and radii. This seminar will discuss different approaches to solving this problem, including some new results based on spectral methods. A new version of the problem will also be presented that uses the masses and tidal deformabilities of neutron stars as the observables (which should soon be measurable with gravitational wave observations).