One-Frequency Schrödinger operators give one of the simplest models where fast transport and localization phenomena are possible. From a dynamical perspective, they can be studied in terms of certain one-parameter families of quasi-periodic co-cycles, which are similarly distinguished as simplest classes of dynamical systems compatible with both KAM phenomena and non-uniform hyperbolicity (NUH). While much studied since the 1970's, until recently the analysis was mostly confined to ''local theories'' describing the KAM and the NUH regimes in detail. In this talk we will describe some of the main aspects of the global theory that has been developed in the last few years.