Feynman integrals are central to the calculation of scattering amplitudes both in particle and gravitational wave physics. This thesis presents advancements in both the analytical and algebraic structure of these integrals and shows how this can be used for efficient evaluation of these integrals. Paper I. In this paper the focus is on one-loop integrals. The singularities of these integrals are fully described and used to derive the full symbol alphabet and canonical differential equation for any number of external particles. It is proven that a large family of one-loop integrals satisfy the Cohen-Macaulay property. Paper II. Two infinite families of Feynman integrals satisfying the Cohen-Macaulay property are classified. This property implies that both the singularities and the number of master integrals is independent of space-time dimension and propagator powers. Paper III. In this paper the singular locus of a Feynman integral is defined as the critical points of a Whitney stratified map. Explicit code and calculations are provided which show that this method captures singularities otherwise hard to detect. Paper IV-V. The algebraic properties of the integrand, especially that of its Newton polytope being a generalized permutohedron, is leverage together with tropical sampling to provide efficient numerical evaluation of Feynman integrals with physical kinematics. The connection between the generalized permutohedron property and the Cohen-Macaulay property is also discussed.