The goal of this note is to give an introduction to locally conformally symplectic and Kähler geometry. In particular, the first two sections aim to provide the reader with enough mathematical background to appreciate these geometric structures. The standard reference for locally conformally Kähler geometry is the book Locally conformal Kähler geometry by Sorin Dragomir and Liviu Ornea; many progresses in this area, however, were accomplished after its publication, hence are not covered there. On the other hand, there is no comprehensive reference for locally conformally symplectic geometry and many recent advances lie scattered in the literature. While the tone of this note is rather expository, I propose a (hopefully) exhaustive bibliography, to which the reader is referred for both the precise statements and the techniques used. Section 3 would like to demonstrate how these geometries can be used to give precise mathematical formulations to ideas deeply rooted in classical and modern Physics.