This book deals with the techniques to solve elliptic eigenvalue problems on polygonal domains using least squares methods. It provides an in-depth treatment of an exponentially accurate nonconforming h-p spectral element method. The spectral element functions are nonconforming and a correction is made to the approximate solution so that the corrected solution is conforming. In this book we discuss the method separately when the boundary of underlying domain is smooth and also when the boundary contains corners. We present error estimates for the eigenvalues and eigenvectors and show that the error in the numerical solution is exponentially small. Theoretical results presented in the book are equally well suited for multiple or clustered eigenvalues.