Computationally solving eigenvalue problems is a central problem in numerical analysis and as such has been the subject of extensive study. In this thesis we present four different methods to compute eigenvalues, each with its own characteristics, strengths and weaknesses. After formally introducing the methods we use them in various numerical experiments to test speed of convergence, stability as well as performance when used to compute eigenfaces, denoise images and compute the eigenvector centrality measure of a graph.