Duality theorems are at the heart of class field theory both for number fields and geometric objects like curves and abelian varieties. They relate abelian Galois extensions with invariants of the base object. In particular, class groups of rings of integers and group schemes attached to Jacobians of curves are involved in this game. Since these groups are the most popular for producing crypto primitives based on discrete logarithms (which use a priori only the Z-linear structure) they carry unavoidably a bilinear structure. In the first lecture we want to sketch the mathematical background and destructive and constructive consequences.