Higher-order gravities are extensions of General Relativity (GR) which introduce in the classical action terms of higher order in the curvature. These appear naturally in the context of string effective actions, as well as in generic gravitational EFTs. Among the myriads of higher-order gravities one may think of, in this talk I will focus on specific theories whose equations of motion are of second order in derivatives for some particular backgrounds. I will provide many examples of such theories, classification results and argue that a subclass of them actually forms a basis for the space of gravitational EFTs. Afterwards, I will discuss the addition of non-minimal couplings to matter, show that some of these theories possess fully regular black hole solutions and conclude with some holographic applications.