  We consider the role of boundary conditions in the $AdS_{d+1}/CFT_{d}$
correspondence for the scalar field theory. Also a careful analysis of some
limiting cases is presented. We study three possible types of boundary
conditions, Dirichlet, Neumann and mixed. We compute the two-point functions of
the conformal operators on the boundary for each type of boundary condition. We
show how particular choices of the mass require different treatments. In the
Dirichlet case we find that there is no double zero in the two-point function
of the operator with conformal dimension $\frac{d}{2}$. The Neumann case leads
to new normalizations for the boundary two-point functions. In the massless
case we show that the conformal dimension of the boundary conformal operator is
precisely the unitarity bound for scalar operators. We find a one-parameter
family of boundary conditions in the mixed case. There are again new
normalizations for the boundary two-point functions. For a particular choice of
the mixed boundary condition and with the mass squared in the range
$-d^2/4<m^2<-d^2/4+1$ the boundary operator has conformal dimension comprised
in the interval $[\frac{d-2}{2}, \frac{d}{2}]$. For mass squared $m^2>-d^2/4+1$
the same choice of mixed boundary condition leads to a boundary operator whose
conformal dimension is the unitarity bound.
