Feynman integrals are special functions with rich hidden structure: large families satisfy linear relations, finite-rank differential systems, and tightly constrained singular behaviour. This talk surveys frameworks that make these features explicit and computationally useful. Topics include twisted period representations and cohomological perspectives on integral relations, D-module methods for organising differential equations, and commutative-algebra tools for identifying the singular locus (Landau singularities). The emphasis will be on intuition and a few illustrative examples, with brief pointers to ongoing applications in multi-scale amplitude computations.