Abstract: If a hexagon is inscribed in a conic, then the three points obtained by intersecting the opposite sides lie on a line. This is Pascal's theorem, first observed in 1639. By considering various pairs of sides obtained from the same six vertices, one obtains a collection of 60 such lines. This collection forms a highly intricate and symmetrical structure, usually called the 'hexagrammum mysticum'. I will explain some of the (myriad) properties of this structure, and the role of algebraic geometry and classical invariant theory in it. The pre-requisites will be kept rather modest, so as to ensure that the talk is widely accessible.