Kirchhoff (Symanzik) polynomials are obtained from a graph as a sum of monomials corresponding to (non-)spanning trees. They are of particular importance in physics in the case of Feynman graphs. One can consider them as special cases of matroid (basis) polynomials or of configuration polynomials. However the latter two generalizations differ in case of non-regular matroids. This more general point of view has the advantage that the classes of matroids and configurations are stable under additional operations such as duality and truncation. In addition there are important configuration polynomials, such as the second graph polynomial, which are not of Kirchhoff type. I will give an introduction to the topic and present new results relating the algebro-geometric structure of the singular locus of configuration hypersurfaces to the structure of the underlying matroid. Their proofs make essential use of matroid theory and, in particular, rely on duality.