Higher-curvature gravities are theories of gravity including terms of higher order in the space-time curvature. They arise naturally when considering an effective approach to gravity. In this talk, I will focus on higher-curvature gravities satisfying two properties: the differential order of their equations of motion gets reduced when restricted to certain specific backgrounds and form a basis for the space of effective theories of gravity. First, I will consider those higher-curvature gravities whose equations of motion become second-order (or less) on top of static and spherically symmetric backgrounds. Secondly, I will focus on those theories whose equations of motion are second-order in derivatives for FLRW configurations, as well as the intersection of the previous two classes of theories. Finally, I will discuss the addition of a non-minimally coupled U(1) gauge field into higher-curvature gravities with simple static and spherically symmetric solutions.