  Degenerate Higher Order Scalar Tensor (DHOST) theories are the most general
scalar-tensor theories whose Lagrangian depends on the metric tensor and a
single scalar field and which propagate only one scalar degree of freedom,
without being plagued by Ostrogradsky instabilities. This is achieved through
certain degeneracies of the functions forming their Lagrangian. They generalise
the Horndeski and beyond-Horndeski theories. Originally build to describe the
late-time acceleration of the expansion of the universe, generalising the
cosmological constant, they can be used to build models for early universe, to
describe inflation or alternatives to standard inflation. In the late universe,
they modify the standard Vainstein screening mechanism from Horndeski theories
which can have observable consequences and are suited to build black hole
models, featuring non-stealth Kerr black hole solutions. In this work we review
their effects on cosmology, looking at their basic properties, their
parameterisations and classifications, and then focusing on solutions in the
early and the late universe.
