In this thesis, we study the spectrum of Schrödinger operators with complex potentials and Dirichlet Laplace operators on domains with rough boundaries. The focus is on spectral approximation results and a-priori bounds for the location and distribution of eigenvalues. Chapter 1 provides an overview of our main results and Chapters 2 - 5 are based on the papers 130, 129, 79, 121 respectively. In Chapter 2, spectral inclusion and pollution results are proved for sequences of linear operators of the form T0+iγsn on a Hilbert space, where sn is strongly convergent to the identity operator and γ > 0. We work in both an abstract setting and a more concrete Sturm-Liouville framework. The results provide rigorous justification for a method of computing eigenvalues in spectral gaps. In Chapter 3, we consider Schrödinger operators of the form HR = −d 2/dx 2 +q+iγχ0,R for large R > 0, where q ∈ L 1 (0,∞) and γ > 0. Bounds for the maximum magnitude of an eigenvalue and for the number of eigenvalues are proved. These bounds complement existing general bounds applied to this operator, for sufficiently large R. In Chapter 4, we prove upper and lower bounds for sums of eigenvalues of Lieb- Thirring type for non-self-adjoint Schrödinger operators on the half-line. The upper bounds are established for general classes of integrable potentials and are shown to be optimal in various senses by proving the lower bounds for specific potentials. We consider sums that correspond to both the critical and non-critical cases. In Chapter 5, we prove a Mosco convergence theorem for H 1 0 spaces of bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counter example showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.