Abstract: We classify higher-curvature gravities whose equations of motion are of second order in derivatives when evaluated in particular ansätze. We commence by considering theories with this particularity in the case of static and spherically symmetric solutions characterized by a single metric function in D dimensions. These theories, dubbed Generalized Quasitopological Gravities (GQTGs), have been extensively studied in the literature given their interesting properties. Here, we present a full classification of all inequivalent GQTGs at all curvature orders n. This is achieved after the explicit construction of a dictionary that allows the uplift of on-shell expressions into fully covariant ones. On the other hand, we consider theories with second order equations of motion when evaluated on Friedmann-Lemaître–Robertson–Walker configurations. We name these theories Cosmological Gravities and initiate a systematic exploration in dimensions D≥3. First, we derive an instance of Cosmological Gravity to all curvature orders and dimensions D≥3. Second, we study Cosmological GQTGs, presenting explicit instances of such theories for all curvature orders and dimensions D≥4. Finally, we investigate the equations of motion for scalar cosmological perturbations, showing that any Cosmological Gravity in D≥3 contains no more than two time derivatives.