  Quantum theory predicts probabilities as well as relative phases between
different alternatives of the system. A unified description of both
probabilities and phases comes through a generalisation of the notion of a
density matrix for histories; this object is the decoherence functional of the
consistent histories approach. If we take phases as well as probabilities as
primitive elements of our theory, we abandon Kolmogorov probability and can
describe quantum theory in terms of fundamental commutative observables,
without being obstructed by Bell's and related theorems. Generalising the
theory of stochastic processes, we develop the description of relative phases
and probabilities for paths on the classical phase space. This description
provides a theory of quantum processes. We identify a number of basic
postulates and study its corresponding properties. We strongly emphasise the
notion of conditioning and are able to write ``quantum differential equations''
as analogous to stochastic differential equations. These can be interpreted as
referring to individual systems. We, then, show the sense in which quantum
theory is equivalent to a quantum process on phase space (using coherent
states). Conversely, starting from quantum processes on phase space we recover
standard quantum theory on Hilbert space from the requirement that the process
satisfies (the Markov property together with time reversibility. The
statistical predictions of our theory are identical to the ones of standard
quantum theory, but the ``logic'' of events is Boolean; events are not
represented by projectors any more. We discuss some implication of this fact
for the interpretation of quantum theory, emphasising that it makes plausible
the existence of realist theories for individual quantum systems.
