40 years ago John Mckay noticed that 196884=196883+1, and Monstrous Moonshine was born. In 1998 Borcherds won a Fields Medal primarily for his work explaining McKay's equation. 5 years ago string theorists Eguchi, Ooguri and Tachikawa observed that 90=45+45,
462=231+231 and 1540=770+770, and Mathieu Moonshine was born. These new equations have still not been explained. My talk will introduce you to these two moonshines.