This book collects an extended version of the lectures delivered by the authors at the Fall Workshop on Geometry and Physics in the years 2014, 2015, 2016.

It aims at introducing advanced graduate and PhD students, as well as young researchers, to current research in mathematics and physics. In particular, it fills the gap between the more physical-oriented and the more mathematical-oriented literature on quantum theory. It introduces various approaches to methods of quantization, along with their impact on modern mathematical methods.