  We establish analogues of the Hawking and Penrose singularity theorems based
on (a) averaged energy conditions with exponential damping; (b) conditions on
local stress-energy averages inspired by the Quantum Energy Inequalities
satisfied by a number of quantum field theories. As particular applications, we
establish singularity theorems for the Einstein equations coupled to a
classical scalar field, which violates the strong energy condition, and the
nonminimally coupled scalar field, which also violates the null energy
condition.
