  We review how the (absence of) Ostrogradsky instability manifests itself in
theories with multiple fields. It has recently been appreciated that when
multiple fields are present, the existence of higher derivatives may not
automatically imply the existence of ghosts. We discuss the connection with
gravitational theories like massive gravity and beyond Horndeski which manifest
higher derivatives in some formulations and yet are free of Ostrogradsky ghost.
We also examine an interesting new class of Extended Scalar-Tensor Theories of
gravity which has been recently proposed. We show that for a subclass of these
theories, the tensor modes are either not dynamical or are infinitely strongly
coupled. Among the remaining theories for which the tensor modes are
well-defined one counts one new model that is not field-redefinable to
Horndeski via a conformal and disformal transformation but that does require
the vacuum to break Lorentz invariance. We discuss the implications for the
effective field theory of dark energy and the stability of the theory.
