  We construct a family of canonical connections and surrounding basic theory
for almost complex manifolds that are equipped with an affine connection. This
framework provides a uniform approach to treating a range of geometries. In
particular we are able to construct an invariant and efficient calculus for
conformal almost Hermitian geometries, and also for almost complex structures
that are equipped with a projective structure. In the latter case we find a
projectively invariant tensor the vanishing of which is necessary and
sufficient for the existence of an almost complex connection compatible with
the path structure. In both the conformal and projective setting we give
torsion characterisations of the canonical connections and introduce certain
interesting higher order invariants.
