  We study various aspects of higher-curvature theories of gravity built from
contractions of the metric, the Riemann tensor and the covariant derivative,
$\mathcal{L}(g^{ab},R_{abcd},\nabla_a)$. We characterise the linearized
spectrum of these theories and compute the modified Newton potential in the
general case. Then, we present the first examples of Generalized
Quasi-topological (GQT) gravities involving covariant derivatives of the
Riemann tensor. We argue that they always have second-order equations on
maximally symmetric backgrounds. Focusing on four spacetime dimensions, we find
new densities of that type involving eight and ten derivatives of the metric.
In the latter case, we find new modifications of the Schwarzschild black hole.
These display thermodynamic properties which depart from the ones of polynomial
GQT black holes. In particular, the relation between the temperature and the
mass of small black holes, $T\sim M^{1/3}$, which universally holds for general
polynomial GQT modifications of Einstein gravity, gets modified in the presence
of the new density with covariant derivatives to $T\sim M^{3}$. Finally, we
consider brane-world gravities induced by Einstein gravity in the AdS bulk. We
show that the effective quadratic action for the brane-world theory involving
arbitrary high-order terms in the action can be written explicitly in a closed
form in terms of Bessel functions. We use this result to compute the propagator
of metric perturbations on the brane and its pole structure in various
dimensions, always finding infinite towers of ghost modes, as well as tachyons
and more exotic modes in some cases.
