  We describe in more detail our approach to the conformal bootstrap which uses
the Mellin representation of $CFT_d$ four point functions and expands them in
terms of crossing symmetric combinations of $AdS_{d+1}$ Witten exchange
functions. We consider arbitrary external scalar operators and set up the
conditions for consistency with the operator product expansion. Namely, we
demand cancellation of spurious powers (of the cross ratios, in position space)
which translate into spurious poles in Mellin space. We discuss two contexts in
which we can immediately apply this method by imposing the simplest set of
constraint equations. The first is the epsilon expansion. We mostly focus on
the Wilson-Fisher fixed point as studied in an epsilon expansion about $d=4$.
We reproduce Feynman diagram results for operator dimensions to $O(\epsilon^3)$
rather straightforwardly. This approach also yields new analytic predictions
for OPE coefficients to the same order which fit nicely with recent numerical
estimates for the Ising model (at $\epsilon =1$). We will also mention some
leading order results for scalar theories near three and six dimensions. The
second context is a large spin expansion, in any dimension, where we are able
to reproduce and go a bit beyond some of the results recently obtained using
the (double) light cone expansion. We also have a preliminary discussion about
numerical implementation of the above bootstrap scheme in the absence of a
small parameter.
