Feynman integrals are at the computational core of quantum field theory, yet despite over 70 years of rich experience, it is probably fair to say that a general theory for their evaluation is lacking. The talk will describe certain steps towards this goal. We shall consider a Feynman diagram of fixed topology (a graph) as a function of all possible parameters (kinematical invariants, masses, numerators, spacetime dimension) and associate with it a system of partial differential equations. The system is associated with a continuous group which is naturally associated with the graph, and its orbits foliate the parameter space. The equation system allows to reduce the diagram to a line integral over simpler diagrams: diagrams gotten by contracting a propagator.