I will introduce the theory of massive gravity proposed by de Rham, Gabadadze and Tolley (2011). I will discuss why this theory could of interest for cosmology and describe some of its unusual observational consequences. I will then go on to show that quantum corrections to the theory, coming from the interactions between the graviton and matter fields, mean that the theory cannot be valid both on cosmological scales and in the laboratory.
Whilst my talk will focus on massive gravity, I will try to place this in the context of more general attempts to build theories of dark energy and modified gravity and discuss why the problems encountered by massive gravity are common to all of these types of theories.