When Newton thought about the moon he always got a headache. The moon is a complicated dynamical system since it is not just attracted by the earth but on a similar scale by the sun. In their endeavour to predict lunar and solar ellipses the Babylonians knew already about three months, the synodic, the anomalistic and the draconitic one. I will explain in the talk how they are related to Floquet multipliers and about their importance in the construction of global surfaces of section for the restricted three-body problem using holomorphic curves as well as their role in space mission design