This dissertation introduces a cohesive framework for numerically computing spectral properties related to the discrete and continuous spectrum of infinite-dimensional operators. Approximations to eigenvalues and eigenvectors, spectral measures, and generalized eigenvectors are constructed by sampling the range of the resolvent operator at strategic points in the complex plane. These algorithms are developed and analyzed directly in the abstract infinite-dimensional Hilbert space setting. They require only two essential computational ingredients: (1) solving linear equations with complex shifts and (2) taking inner products in the Hilbert space. Numerical implementations for a broad class of differential and integral operators, leveraging state-of-the-art adaptive spectral methods, are provided in an accompanying MATLAB package called SpecSolve, which is demonstrated through a collection of examples.