The McKay conjecture is the origin of the counting conjectures in the representation theory of finite groups. This book gives a comprehensive introduction to these conjectures, while assuming minimal background knowledge. Character theory is explored in detail along the way, from the very basics to the state of the art. This includes not only older theorems, but some brand new ones too. New, elegant proofs bring the reader up to date on progress in the field, leading to the final proof that if all Finite simple groups satisfy the inductive McKay condition, then the McKay conjecture is true. Open questions are presented throughout the book, and each chapter ends with a list of problems, with varying degrees of difficulty.