This is an almost verbatim reproduction of the author's lecture notes written in 198384 at Ohio State University, Columbus. A substantial update is given in the bibliography. Over the last 20 plus years there has been energetic activity in the field of finite simple group theory related to the monster simple group. Most notably, influential works have been produced in the theory of vertex operator algebras from research that was stimulated by the moonshine of the finite groups. Still, we can ask the same questions now that we did 3040 years ago: What is the monster simple group? Is it really related to the theory of the universe as it was vaguely so envisioned? What lies behind the moonshine phenomena of the monster group? It may appear that we have only scratched the surface. These notes are primarily reproduced for the benefit of readers who wish to start learning about modular functions used in moonshine. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Undergraduate and graduate students interested in modular functions used in moonshine. Table of Contents  Modular functions and modular forms
 Dedekind eta function
 "Moonshine" of finite groups
 Multiplicative product of \(\eta\) functions
 Appendix. Genus zero discrete groups
 Bibliography
