In this talk I will discuss two computational algebraic problems associated with Feynman integrals. The first concerns the determination of Landau singularities, viewed as a discriminant locus in parameter space where the topology of the underlying geometric setup changes. In ongoing work, we study this locus through the analysis of critical points and relate it to jumps in the Euler characteristic. The second concerns the computation of annihilating differential operators and the corresponding D-module of a Feynman integral. I will describe an algorithmic approach based on a twisted version of Griffiths–Dwork reduction, which allows one to derive the relevant D-ideal directly from the integral representation. The aim of the talk is to illustrate how questions about singularities and differential equations of Feynman integrals can be formulated and attacked using tools from computational algebraic geometry.