Even apparently simple dynamical systems as those described by maps of an interval can display a rich variety of different asymptotic behaviors. On a measure theoretic level these types of behaviour are described by invariant measures, i.e. measures describing statistically stationary states of the system. Not all of them are of equal importance, but the "observable" ones deserve special attention. In the first part of my talk I describe these concepts in simple cases (expanding maps) and relate the property of a measure of being observable to other ergodic theoretic properties. In the second part I discuss the possibility to extend the notions and results from the first part to maps like the logistic map (and other unimodal maps).