  We review the consistent histories formulations of quantum mechanics
developed by Griffiths, Omn\`es and Gell-Mann and Hartle, and describe the
classification of consistent sets. We illustrate some general features of
consistent sets by a few simple lemmas and examples. We consider various
interpretations of the formalism, and examine the new problems which arise in
reconstructing the past and predicting the future. It is shown that Omn\`es'
characterisation of true statements --- statements which can be deduced
unconditionally in his interpretation --- is incorrect. We examine critically
Gell-Mann and Hartle's interpretation of the formalism, and in particular their
discussions of communication, prediction and retrodiction, and conclude that
their explanation of the apparent persistence of quasiclassicality relies on
assumptions about an as yet unknown theory of experience. Our overall
conclusion is that the consistent histories approach illustrates the need to
supplement quantum mechanics by some selection principle in order to produce a
fundamental theory capable of unconditional predictions.
