Kurzbeschreibung: An "open dynamical system” is obtained by cutting a “hole” in the phase space of a dynamical system, and considering the set of points that never falls into the hole under forward iteration. As a simple, but already nontrivial, example, one can consider the set of points that never hit a fixed interval under iteration of the doubling map. It is of interest to look at the Hausdorff dimension of the set of remaining points (sometimes called the survival set), and to consider natural measures, such as Gibbs measures, supported on it. If the hole varies in a family (for instance, if one removes the interval (0, t) as t varies), all the associated quantities vary with the parameter, and one looks at how fast they vary, looking for example at their Holder exponent. In joint work with T. Das, M. Urbanski, and A. Zdunik, we provide a general framework to deal with thermodynamic formalism of open dynamical systems in metric spaces, and establish formulas for escape rates in this general context.