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This talk provides a self-contained, fairly elementary introduction to moonshine. I first review the developments of the 1980s and 1990s, demonstrating a striking and unexpected deep relationship between the largest of the sporadic simple groups (the Monster), basic objects in the theory of modular forms, and string theory. These mysterious relations came to be called “moonshine.” I then describe the reason for renewed interest in moonshine: the emergence of Mathieu and Umbral moonshines in studies of canonical superstring theory compactifications, such as that involving the K3 surface. I close with a description of some ongoing work attempting to provide a physical underpinning for Umbral moonshine by using facts about S-duality symmetries of string theory in three dimensions.