In 2010, Zagier defined the notion of a "quantum modular form,'' and offered several diverse examples, including Kontsevich's 'strange' function. Here, we construct infinite families of quantum modular forms, and prove one of Ramanujan's remaining claims about mock theta functions in his last letter to Hardy as a special case of our work. We will show how quantum modular forms underlie new relationships between combinatorial mock modular and modular forms due to Dyson and Andrews-Garvan. This is joint work with Ken Ono (Emory U.) and Rob Rhoades (CCR-Princeton).