  In analogy with the classical theory of Eichler integrals for integral weight
modular forms, Lawrence and Zagier considered examples of Eichler integrals of
certain half-integral weight modular forms. These served as early prototypes of
a new type of object, which Zagier later called a quantum modular form. Since
then, a number of others have studied similar examples. Here we develop the
theory in a general context, giving rise to a well-defined class of quantum
modular forms. Since elements of this class show up frequently in examples of
combinatorial and number theoretical interest, we propose the study of the
general properties of this space of quantum modular forms. We conclude by
giving a set of fundamental questions concerning this space of objects which
merit further study.
