We give a pedagogical introduction to the concept of martingales: The contribution addresses especially students participating in the International Conference "Let's Face Chaos Through Nonlinear Dynamics" in Maribor 2002. We deal in particular with Doob's Upcrossing Lemma and present some applications. A recently established connection between the structure of martingales and Hermite functions by Fitzsimmons is presented. Basic properties of the classical Hermite functions are revised. There remains the question whether one is able to generalize the obtained results in context of a discrete scenario, i.e. when replacing the used continuous Hermite functions by suitable discrete ones.