  Polarized and unpolarized neutron diffraction has been used to search for
magnetic order in YBa$_2$Cu$_3$O$_{6+x}$ superconductors. Most of the
measurements were made on a high quality crystal of YBa$_2$Cu$_3$O$_{6.6}$. It
is shown that this crystal has highly ordered ortho-II chain order, and a sharp
superconducting transition. Inelastic scattering measurements display a very
clean spin-gap and pseudogap with any intensity at 10 meV being 50 times
smaller than the resonance intensity. The crystal shows a complicated magnetic
order that appears to have three components. A magnetic phase is found at high
temperatures that seems to stem from an impurity with a moment that is in the
$a$-$b$ plane, but disordered on the crystal lattice. A second ordering occurs
near the pseudogap temperature that has a shorter correlation length than the
high temperature phase and a moment direction that is at least partly along the
c-axis of the crystal. Its moment direction, temperature dependence, and Bragg
intensities suggest that it may stem from orbital ordering of the $d$-density
wave (DDW) type. An additional intensity increase occurs below the
superconducting transition. The magnetic intensity in these phases does not
change noticeably in a 7 Tesla magnetic field aligned approximately along the
c-axis. Searches for magnetic order in YBa$_2$Cu$_3$O$_{7}$ show no signal
while a small magnetic intensity is found in YBa$_2$Cu$_3$O$_{6.45}$ that is
consistent with c-axis directed magnetic order. The results are contrasted with
other recent neutron measurements.
