The paper determines the most general scalar field theories that have actions depending on derivatives of order two or less and equations of motion that remain second order or lower in flat spacetime. It shows that these theories can be constructed from linear combinations of Lagrangians formed by multiplying a specific form of the Galileon Lagrangian by an arbitrary scalar function of the scalar field and its first derivatives. The authors also derive curved spacetime extensions of these theories, maintaining second-order field equations for both the metric and the scalar field. This provides the most general extension of k-essence, Galileons, k-mouflage, and kinetically braided scalars under the condition that field equations remain second order. Additionally, they derive the most general action for a scalar classicalizer with second-order field equations. The paper discusses the relation between their construction and the Euler hierarchies of Fairlie et al., showing that Euler hierarchies allow the construction of the most general theory when it is shift symmetric. As an application, they provide the covariantized version of the conformal Galileon.The paper determines the most general scalar field theories that have actions depending on derivatives of order two or less and equations of motion that remain second order or lower in flat spacetime. It shows that these theories can be constructed from linear combinations of Lagrangians formed by multiplying a specific form of the Galileon Lagrangian by an arbitrary scalar function of the scalar field and its first derivatives. The authors also derive curved spacetime extensions of these theories, maintaining second-order field equations for both the metric and the scalar field. This provides the most general extension of k-essence, Galileons, k-mouflage, and kinetically braided scalars under the condition that field equations remain second order. Additionally, they derive the most general action for a scalar classicalizer with second-order field equations. The paper discusses the relation between their construction and the Euler hierarchies of Fairlie et al., showing that Euler hierarchies allow the construction of the most general theory when it is shift symmetric. As an application, they provide the covariantized version of the conformal Galileon.