Among the allowed representations of extended supersymmetry in six dimensions there are exotic chiral multiplets that, instead of a graviton, contain mixed-symmetry spin-$2$ tensor fields. Notably, an $\mathcal{N}=(4,0)$ multiplet has a four index exotic graviton and it was conjectured that an interacting theory based on this multiplet could arise as a strong coupling limit of M theory compactified on $T^6$. We present an algebraic study of these multiplets and their possible embedding into the framework of exceptional field theory, finding in particular that the six-dimensional momenta do not correspond to a conventional spacetime section. When compactified on a circle, the six-dimensional multiplets give rise to the same degrees of freedom as five-dimensional supergravity theories with the same number of supersymmetries. However, by considering anomalies and the generation of Chern-Simons couplings, we find reason to doubt that their dynamics will agree with the five-dimensional gravity theories. We propose an alternative picture, similar to F-theory, in which particular fixed-volume $T^3$-fibered spacetimes play a central role, suggesting that only on compactification to three-dimensions will one make contact with the dynamics of supergravity. In these exotic multiplets, there are also rank two antisymmetric tensor-spinors. In the last part of the thesis, we perform the quantisation of general antisymmetric tensor-spinors $\psi^\alpha_{\mu_1 \dots \mu_p}$ using the Batalin-Vilkovisky field-antifield formalism for any $p$ and in arbitrary dimensions. Just as for the gravitino ($p=1$), an extra propagating Nielsen-Kallosh ghost appears in quadratic gauges containing a differential operator. The appearance of this ``third ghost" is described within the BV formalism for arbitrary reducible gauge theories. We then use the resulting spectrum of ghosts and the Atiyah-Singer index theorem to compute gravitational anomalies.