Measures that do not fit into the classical well-established framework of Gibbs measures arise when a system acquires an infinite-range dependence.  The best-known examples for this to happen are the so-called renormalization group pathologies.  A particular issue is the emergence of non-Gibbsianness under stochastic time-evolutions, starting from an initial Gibbs-measure.  We will present general results, describe the analysis of solvable model systems, and discuss the "multiple-history"- mechanism for the emergence of
non-Gibbsianness.