  We determine the most general scalar field theories which have an action that
depends on derivatives of order two or less, and have equations of motion that
stay second order and lower on flat space-time. We show that those theories can
all be obtained from linear combinations of Lagrangians made by multiplying a
particular form of the Galileon Lagrangian by an arbitrary scalar function of
the scalar field and its first derivatives. We also obtain curved space-time
extensions of those theories which have second order field equations for both
the metric and the scalar field. This provide the most general extension, under
the condition that field equations stay second order, of k-essence, Galileons,
k-Mouflage as well as of the kinetically braided scalars. It also gives the
most general action for a scalar classicalizer, which has second order field
equations. We discuss the relation between our construction and the Euler
hierachies of Fairlie et al, showing in particular that Euler hierachies allow
one to obtain the most general theory when the latter is shift symmetric. As a
simple application of our formalism, we give the covariantized version of the
conformal Galileon.
