We review the consistent histories formulations of quantum mechanics developed by Griffiths, Omn6s, 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 Omnes' characterization 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.