The theory of mock modular forms is fairly new, having been initiated formally only in 2002, but is motivated by the 17 “mock theta functions” that Ramanujan described in a letter to Hardy much earlier, in 1920. Since 2002 these objects have occurred in many places in both mathematics and mathematical physics, notably in the string theory of black holes and in the so-called “Mathieu” and “umbral” moonshine, and they also occur in connection with quantum invariants of knots and 3-dimensional manifolds. These two talks, which assume no prerequisites, will give an introduction to the suject and some of its applications.