The discriminant of a polynomial with prescribed monomials is an irreducible polynomial in the coefficients vanishing when the corresponding polynomial has multiple roots. It is an important tool in numerical studies of polynomials and describes a rich geometry of tangency of an associated algebraic variety. I will recall this interplay and introduce the notion of discriminant for a system of polynomials (the mixed discriminant). I will then show how a generalised Schäfli method leading to the notion of iterated discriminant can simplify the analysis of mixed discriminants. The talk will mostly be introductory to the subject. The latest results are based on joint work with Alicia Dickenstein and Ralph Morrison.