One of the biggest open problems in physics is the consistent
unification of quantum theory with general relativity,
resulting in a theory of "quantum gravity".
Such a theory would have an important bearing upon the physics
of the early universe, the understanding of black holes,
and the structure of spacetime.
In my talk I give a general introduction to the motivations for
and the problems of quantum gravity. I shall then briefly describe
the main approaches -- quantum general relativity and
string theory -- and their applications.
I shall present an overview of recent results and current research,
including such issues as the microstructure of space,
black-hole entropy, and quantum cosmology.