I review recent work towards the construction of exotic supergravity theories in six dimensions with maximal (4,0) and (3,1) supersymmetry. These theories were conjectured by C. Hull to exist but so far have only been constructed at the free level. The interacting theories must be exotic or non-geometric since they feature mixed symmetry tensors instead of a conventional graviton, hence suggesting the need for a generalized notion of spacetime and diffeomorphism invariance.