  We review the most general scalar-tensor cosmological models with up to
second-order derivatives in the field equations that have a fixed spatially
flat de Sitter critical point independent of the material content or vacuum
energy. This subclass of the Horndeski Lagrangian is capable of dynamically
adjusting any value of the vacuum energy of the matter fields at the critical
point. We present the cosmological evolution of the linear models and the
non-linear models with shift symmetry. We come to the conclusion that the shift
symmetric non-linear models can deliver a viable background compatible with
current observations.
