Abstract. Lovelock and Horndeski theories are natural generalisations of Einstein's theory of General Relativity with second order equations of motion.

While their applications in astrophysics and cosmology have been thoroughly explored, the issue of their mathematical consistency has received little attention.

In particular, it is not known whether these theories admit a well-posed initial value problem, a necessary requirement for any classical theory to make sense. If this condition were not satisfied, these theories would not constitute a viable alternative to General Relativity.

In this talk, I will discuss why the standard method used to establish the local well-posedness of the Einstein equations fails for these theories.