A key problem in Arithmetic Geometry is to find and describe the solutions of polynomial equations in number fields like Q. This very difficult problem can be solved completely only in very special cases. Research has therefore turned to qualitative statements like the Birch-Swinnerton-Dyer conjecture for elliptic curves. This conjectural formula from 1960s relates the solutions of the equations defining an elliptic curve to the special value of its L-function and can be seen as a very sophisticated local-global principle. Much research was devoted to this conjecture and many important concepts in number theory were invented to tackle it. Nowadays, the Birch-Swinnerton-Dyer conjecture is a special case of the Bloch-Kato conjecture, which predicts similar formulas for all varieties over number fields. In this talk we will give a gentle introduction to the Birch-Swinnerton-Dyer-conjecture for general mathematical audience. We will also survey some of the results obtained in the direction of this conjecture. This talk is part of the "Millenniumsfestival" . There will be a livestream available on the "YouTube channel Mathematic Münster".