In his final letter to Hardy in 1920, Ramanujan described several q-series he called "mock theta functions", which he thought "should enter into mathematics as beautifully as ordinary theta functions". Understanding these functions and their relationship to classical modular forms sparked a mathematical mystery that persisted until the invention of mock modular forms and harmonic weak Maass forms in the works of Zwegers (2002) and Bruinier-Funke (2004). This talk will explain the basics of mock modular forms and explore a few applications to combinatorics and arithmetic geometry. If time permits, then we will discuss the open problem of understanding "mock automorphic representations". Speaker(s): Patrick Kelley (UM)