In this paper we elaborate on the structure of the continuous-time histories
description of quantum theory, which stems from the consistent histories
scheme. In particular, we examine the construction of history Hilbert space,
the identification of history observables and the form of the decoherence
functional (the object that contains the probability information). It is shown
how the latter is equivalent to the closed-time-path (CTP) generating
functional. We also study the phase space structure of the theory first through
the construction of general representations of the history group (the analogue
of the Weyl group) and the implementation of a histories Wigner-Weyl transform.
The latter enables us to write quantum theory solely in terms of phase space
quantities. These results enable the implementation of an algorithm for
identifying the classical (stochastic) limit of a general quantum system.