  We revisit the subject of holographic renormalization for asymptotically AdS
spacetimes. For many applications of holography, one has to handle the
divergences associated with the on-shell gravitational action. The brute force
approach uses the Fefferman-Graham (FG) expansion near the AdS boundary to
identify the divergences, but subsequent reversal of the expansion is needed to
construct the infinite counterterms. While in principle straightforward, the
method is cumbersome and application/reversal of FG is formally unsatisfactory.
Various authors have proposed an alternative method based on the
Hamilton-Jacobi equation. However, this approach may appear to be abstract,
difficult to implement, and in some cases limited in applicability. In this
paper, we clarify the Hamilton-Jacobi formulation of holographic
renormalization and present a simple algorithm for its implementation to
extract cleanly the infinite counterterms. While the derivation of the method
relies on the Hamiltonian formulation of general relativity, the actual
application of our algorithm does not. The work applies to any $D$-dimensional
holographic dual with asymptotic AdS boundary, Euclidean or Lorentzian, and
arbitrary slicing. We illustrate the method in several examples, including the
FGPW model, a holographic model of 3d ABJM theory, and cases with marginal
scalars such as a dilaton-axion system.
