  In this thesis I consider the general problem of how to make the best
possible phase measurements using feedback. Both the optimum input state and
optimum feedback are considered for both single-mode dyne measurements and
two-mode interferometric measurements. I derive the optimum input states under
general dyne measurements when the mean photon number is fixed, both for
general states and squeezed states. I propose a new feedback scheme that
introduces far less phase uncertainty than mark II feedback, and is very close
to the theoretical limit. I also derive results for the phase variance when
there is a time delay in the feedback loop, showing that there is a lower limit
to the introduced phase variance, and this is approached quite accurately under
some conditions. I derive the optimum input states for interferometry, showing
that the phase uncertainty scales as 1/N for all the common measures of
uncertainty. This is contrasted with the |j0>_z state, which does not scale as
1/N for all measures of phase uncertainty. I introduce an adaptive feedback
scheme that is very close to optimum, and can give scaling very close to 1/N
for the uncertainty. Lastly I consider the case of continuous measurements, for
both the dyne and interferometric cases.
