We start with a preliminary overview of Sobolev spaces, very useful in the modern approach to solve PDEs. We then introduce an eigenvalue problem arising from time-harmonic Maxwell's equations, deriving its so-called weak formulation and showing some first properties. The last part of the seminar will focus on the domain perturbation, inspecting the dependence of the eigenvalues upon variation of the region in which we set our problem, and presenting a result regarding shape optimization.