  Recently, the identification of new higher-curvature interactions known as
Einsteinian cubic gravity (ECG) and Generalized quasi-topological gravities
(GQGs) has allowed for important advances in the study of black hole solutions
and holographic applications of higher-order gravity. These theories are
characterized by having second-order linearized equations on maximally
symmetric background and by possessing static, spherically symmetric (SSS)
black hole solutions satisfying $g_{tt}g_{rr}=-1$, and whose thermodynamic
properties can be studied analytically in a fully non-perturbative fashion. In
the first part of this thesis, we perform a detailed analysis of the weak-field
limit of $\mathcal{L}($Riemann$)$ theories and of the condition
$g_{tt}g_{rr}=-1$ on SSS solutions, and we review the construction of ECG and
Generalized quasi-topological gravities. In addition, we show that GQGs could
serve as building blocks to construct the most general EFT for gravity. In the
second part we study the non-perturbative corrections to the asymptotically
flat Schwarzschild black hole in $D=4$. We obtain closed exact expressions for
the thermodynamic properties of spherically symmetric black holes with an
arbitrary number of higher-curvature terms. We find that, quite generally, the
higher-curvature corrections make the specific heat of these black holes
positive below certain mass, and hence black holes take an infinite time to
evaporate and the final "Hawking explosion" is avoided. In the last part we
study asymptotically AdS black holes and Euclidean-AdS-Taub-NUT solutions and
we establish various entries of the holographic dictionary of $D=4$ ECG. We
illustrate the power of holographic higher-order gravities by obtaining a
relation for the free energy of CFTs on squashed spheres, which we conjecture
to be valid for arbitrary CFTs.
