We introduce a new class of scalar-tensor theories of gravity that extend Horndeski, or " generalized galileon " , models. Despite possessing equations of motion of higher order in derivatives, we show that the true propagating degrees of freedom obey well-behaved second-order equations and are thus free from Ostrogradski instabilities, in contrast to standard lore. Remarkably, the covariant versions of the original galileon Lagrangians—obtained by direct replacement of derivatives with covariant derivatives—belong to this class of theories. These extensions of Horndeski theories exhibit an uncommon, interesting phenomenology: The scalar degree of freedom affects the speed of sound of matter, even when the latter is minimally coupled to gravity. The discovery of the present cosmological acceleration has spurred the exploration of gravitational theories that could account for this effect. Many extensions of general relativity (GR) are based on the inclusion of a scalar degree of freedom (DOF) in addition to the two ten-sor propagating modes of GR (see e.g. [1] for a review). In this context, a recent important proposal is the so-called galileon models [2], with Lagrangians that involve second-order derivatives of the scalar field and lead, nevertheless , to equations of motions of second order. Such a property guarantees the avoidance of Ostrogradski in-stabilities, i.e. of the ghost-like DOF that are usually associated with higher time derivatives (see e.g. [3]). Initially introduced in Minkowski spacetime, galileons have then been generalized to curved spacetimes [4–6], where they turn out to be equivalent to a class of theories originally constructed by Horndeski forty years ago [7]. Today, Horndeski theories, which include quintessence, k-essence and f (R) models, constitute the main theoretical framework for scalar-tensor theories, in which cosmologi-cal observations are interpreted. The purpose of this Letter is to show that this framework is not as exhaustive as generally believed, and can in fact be extended to include new Lagrangians. Indeed, having equations of motion of second order in derivatives—while indeed sufficient—is not necessary to avoid Ostrogradski instabilities, as already pointed out in e.g. [8, 9]. The theories beyond Horndeski that we propose lead to distinct observational effects and are thus fully relevant for an extensive comparison of scalar-tensor theories with observations.