  This article reviews scalar-tensor theories characterized by a Lagrangian
that, despite the presence of second order derivatives, contain a single scalar
degree of freedom. These theories, known as Degenerate Higher-Order
Scalar-Tensor (DHOST) theories, include Horndeski and Beyond Horndeski
theories. They propagate a single scalar mode as a consequence of the
degeneracy of their Lagrangian and, therefore, are not plagued by an
Ostrodradsky instability. They have been fully classified up to cubic order in
second-order derivatives. The study of their phenomenological consequences
restricts the subclass of DHOST theories that are compatible with observations.
In cosmology, these theories can be described in the language of the unified
effective approach to dark energy and modified gravity. Compact objects in the
context of DHOST theories are also discussed.
