  This review summarizes the current status of the energy conditions in general
relativity and quantum field theory. We provide a historical review and a
summary of technical results and applications, complemented with a few new
derivations and discussions. We pay special attention to the role of the
equations of motion and to the relation between classical and quantum theories.
  Pointwise energy conditions were first introduced as physically reasonable
restrictions on matter in the context of general relativity. They aim to
express e.g. the positivity of mass or the attractiveness of gravity. Perhaps
more importantly, they have been used as assumptions in mathematical relativity
to prove singularity theorems and the non-existence of wormholes and similar
exotic phenomena. However, the delicate balance between conceptual simplicity,
general validity and strong results has faced serious challenges, because all
pointwise energy conditions are systematically violated by quantum fields and
also by some rather simple classical fields. In response to these challenges,
weaker statements were introduced, such as quantum energy inequalities and
averaged energy conditions. These have a larger range of validity and may still
suffice to prove at least some of the earlier results. One of these conditions,
the achronal averaged null energy condition, has recently received increased
attention. It is expected to be a universal property of the dynamics of all
gravitating physical matter, even in the context of semiclassical or quantum
gravity.
