  We present a number of open problems within general relativity. After a brief
introduction to some technical mathematical issues and the famous singularity
theorems, we discuss the cosmic censorship hypothesis and the Penrose
inequality, the uniqueness of black hole solutions and the stability of Kerr
spacetime and the final state conjecture, critical phenomena and the
Einstein-Yang--Mills equations, and a number of other problems in classical
general relativity. We then broaden the scope and discuss some mathematical
problems motivated by quantum gravity, including AdS/CFT correspondence and
problems in higher dimensions and, in particular, the instability of anti-de
Sitter spacetime, and in cosmology, including the cosmological constant problem
and dark energy, the stability of de Sitter spacetime and cosmological
singularities and spikes. Finally, we briefly discuss some problems in
numerical relativity and relativistic astrophysics.
