  Higher-order curvature corrections involving the conformally-invariant
Weyl-squared action have played a role in two recent investigations of
four-dimensional gravity; in critical gravity, where it is added to the
standard cosmological Einstein-Hilbert action with a coefficient tuned to make
the massive ghostlike spin-2 excitations massless, and in a pure Weyl-squared
action considered by Maldacena, where the massive spin-2 modes are removed by
the imposition of boundary conditions. We exhibit the connections between the
two approaches, and we also generalise critical gravity to a wider class of
Weyl-squared modifications to cosmological Einstein gravity where one can
eliminate the massive ghostlike spin-2 modes by means of boundary conditions.
The cosmological constant plays a crucial role in the discussion, since there
is then a "window" of negative mass-squared spin-2 modes around AdS_4 that are
not tachyonic. We also construct analogous conformal and non-conformal
gravities in six dimensions.
