  The goal of this note is to give an introduction to locally conformally
symplectic and K\"ahler 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 K\"ahler 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.
