  We have recently proposed a new class of gravitational scalar-tensor theories
free from Ostrogradski instabilities, in arXiv:1404.6495. As they generalize
Horndeski theories, or "generalized" galileons, we call them G$^3$. These
theories possess a simple formulation when the time hypersurfaces are chosen to
coincide with the uniform scalar field hypersurfaces. We confirm that they
contain only three propagating degrees of freedom by presenting the details of
the Hamiltonian formulation. We examine the coupling between these theories and
matter. Moreover, we investigate how they transform under a disformal
redefinition of the metric. Remarkably, these theories are preserved by
disformal transformations that depend on the scalar field gradient, which also
allow to map subfamilies of G$^3$ into Horndeski theories.
