  We review and discuss some recent progress in Lovelock and Horndeski theories
modifying Einstein's General Relativity. Using as our guide the uniqueness
properties of these modified gravity theories we then discuss how Kaluza-Klein
reduction of Lovelock theory can lead to effective scalar-tensor actions
including several important terms of Horndeski theory. We show how this can be
put to practical use by mapping analytic black hole solutions of one theory to
the other. We then elaborate on the subset of Horndeski theory that has
self-tuning properties and review a generic method giving scalar-tensor black
hole solutions.
