  Conformal theories in a d dimensional spacetime may be expressed as
manifestly conformal theories in a d+2 dimensional conformal space as first
proposed by Dirac. The reduction to d dimensions goes via the d+1 dimensional
hypercone in the conformal space. Here we give a rather extensive expose of
such theories. We review and extend the theory of spinning conformal particles.
We give a precise and geometrical formulation of manifestly conformal fields
for which we give a consistent action principle. The requirement of invariance
under special gauge transformations off the hypercone plays a fundamental role
here. Maxwell's theory and linear conformal gravity are derived in the
conformal space and are treated in detail. Finally, we propose a consistent
coordinate invariant action principle in the conformal space and give an action
that should correspond to conformal gravity.
