Locally conformally symplectic and Kähler geometry

The goal of this note is to give an introduction to locally conformally symplectic and Kähler geometry. In particular, Sections 1 and 3 aim to provide the reader with enough mathematical background to appreciate this kind of geometry. The reference book for locally conformally Kähler geometry is "Locally conformal K"ahler Geometry" by Sorin Dragomir and Liviu Ornea. Many progresses in this field, however, were accomplished after the publication of this book, hence are not contained there. On the other hand, there is no book on locally conformally symplectic geometry and many recent advances lie scattered in the literature. Sections 2 and 4 would like to demonstrate how these geometries can be used to give precise mathematical formulations to ideas deeply rooted in classical and modern Physics.