One of the basic properties of dynamical systems is that local instability of trajectories gives rise to a global “chaotic” behavior. This local instability can be described as some kind of hyperbolicity. Smooth Ergodic Theory investigates the metric and stochastic properties of measures invariant under differentiate mappings or flows on manifolds. The consideration of invariant measures allows to “tame” the “chaotic” behavior from a probabilistic point of view. This transition from differentiable structures to measurable structures and vice versa makes this field fascinating and paves the way to applications far beyond this field. Due to its generality the methods and results of Smooth Ergodic Theory entered areas as Riemannian Geometry, Number Theory, Statistical Physics, Partial Differential Equations or Numerical Simulations.