  The standard formulation of the AdS/CFT correspondence is incomplete since it
requires adding to a supergravity action some a priori unknown boundary terms.
We suggest a modification of the correspondence principle based on the
Hamiltonian formulation of the supergravity action, which does not require any
boundary terms. Then all the boundary terms of the standard formulation
naturally appear by passing from the Hamiltonian version to the Lagrangian one.
As examples the graviton part of the supergravity action on the product of
$AdS_{d+1}$ with a compact Einstein manifold $\cal E$ and fermions on
$AdS_{d+1}$ are considered. We also discuss conformal transformations of
gravity fields on the boundary of $AdS$ and show that they are induced by the
isometries of $AdS$.
