This paper presents a generalization of scalar field theories with actions depending on derivatives of order two or less, and equations of motion that remain second order in flat space-time. The authors show that all such theories can be obtained from linear combinations of Lagrangians constructed by multiplying a specific form of the Galileon Lagrangian by an arbitrary scalar function of the scalar field and its first derivatives. They also derive curved space-time extensions of these theories, which maintain 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. The paper also discusses the relation between their construction and the Euler hierarchies of Fairlie et al., showing that Euler hierarchies allow one to obtain the most general theory when shift symmetry is present. As a simple application, the paper gives the covariantized version of the conformal Galileon. The paper also discusses the uniqueness of these theories in flat space-time, showing that the most general theory satisfying the three conditions (i) its Lagrangian contains derivatives of order two or less of the scalar field, (ii) its Lagrangian is polynomial in the second derivatives of the scalar field, and (iii) the corresponding field equations are of order two or lower in derivatives, has a Lagrangian that is an arbitrary linear combination of the Galileon Lagrangians. The paper also discusses the covariantization of these theories in curved space-time, showing that non-minimal couplings can be introduced to eliminate higher order derivatives. The paper concludes that the most general Lagrangian in D-dimensions obeying these conditions is given by a linear combination of the Galileon Lagrangians.This paper presents a generalization of scalar field theories with actions depending on derivatives of order two or less, and equations of motion that remain second order in flat space-time. The authors show that all such theories can be obtained from linear combinations of Lagrangians constructed by multiplying a specific form of the Galileon Lagrangian by an arbitrary scalar function of the scalar field and its first derivatives. They also derive curved space-time extensions of these theories, which maintain 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. The paper also discusses the relation between their construction and the Euler hierarchies of Fairlie et al., showing that Euler hierarchies allow one to obtain the most general theory when shift symmetry is present. As a simple application, the paper gives the covariantized version of the conformal Galileon. The paper also discusses the uniqueness of these theories in flat space-time, showing that the most general theory satisfying the three conditions (i) its Lagrangian contains derivatives of order two or less of the scalar field, (ii) its Lagrangian is polynomial in the second derivatives of the scalar field, and (iii) the corresponding field equations are of order two or lower in derivatives, has a Lagrangian that is an arbitrary linear combination of the Galileon Lagrangians. The paper also discusses the covariantization of these theories in curved space-time, showing that non-minimal couplings can be introduced to eliminate higher order derivatives. The paper concludes that the most general Lagrangian in D-dimensions obeying these conditions is given by a linear combination of the Galileon Lagrangians.