  In this paper we discuss the dynamics of conformal field theories on anti-de
Sitter space, focussing on the special case of the N=4 supersymmetric
Yang-Mills theory on AdS_4. We argue that the choice of boundary conditions, in
particular for the gauge field, has a large effect on the dynamics. For
example, for weak coupling, one of two natural choices of boundary conditions
for the gauge field leads to a large N deconfinement phase transition as a
function of the temperature, while the other does not. For boundary conditions
that preserve supersymmetry, the strong coupling dynamics can be analyzed using
S-duality (relevant for g_{YM} >> 1), utilizing results of Gaiotto and Witten,
as well as by using the AdS/CFT correspondence (relevant for large N and large
't Hooft coupling). We argue that some very specific choices of boundary
conditions lead to a simple dual gravitational description for this theory,
while for most choices the gravitational dual is not known. In the cases where
the gravitational dual is known, we discuss the phase structure at large 't
Hooft coupling.
