  We investigate the details of the bulk-boundary correspondence in Lorentzian
signature anti-de Sitter space. Operators in the boundary theory couple to
sources identified with the boundary values of non-normalizable bulk modes.
Such modes do not fluctuate and provide classical backgrounds on which bulk
excitations propagate. Normalizable modes in the bulk arise as a set of
saddlepoints of the action for a fixed boundary condition. They fluctuate and
describe the Hilbert space of physical states. We provide an explicit, complete
set of both types of modes for free scalar fields in global and Poincar\'e
coordinates. For $\ads{3}$, the normalizable and non-normalizable modes
originate in the possible representations of the isometry group
$\SL_L\times\SL_R$ for a field of given mass. We discuss the group properties
of mode solutions in both global and Poincar\'e coordinates and their relation
to different expansions of operators on the cylinder and on the plane. Finally,
we discuss the extent to which the boundary theory is a useful description of
the bulk spacetime.
