  Horndeski gravity holds a special position as the most general extension of
Einstein's theory of general relativity with a single scalar degree of freedom
and second-order field equations. Because of these features, Horndeski gravity
is an attractive phenomenological playground to investigate the consequences of
modifications of general relativity in cosmology and astrophysics. We present a
review of the progress made so far in the study of compact objects (black holes
and neutron stars) within Horndeski gravity. In particular, we review our
recent work on slowly rotating black holes and present some new results on
slowly rotating neutron stars.
