  We study generalisations of ghost-free bimetric theory which involve more
than two spin-2 fields. The consistent interactions can enter in the form of
two different couplings and in the majority of this work we concentrate on the
simpler one. The corresponding action involves one metric coupled to $N$ tensor
fields which do not interact with each other. We derive maximally symmetric
solutions to the multimetric equations of motion and identify the mass
eigenstates in the linearised theory around these backgrounds. Our results are
then applied to the problem of singling out multimetric models which possess
certain additional structures. In particular, we look for a relation between
scale invariant background solutions, the perturbative emergence of Weyl
invariance and the presence of partially massless spin-2 fields in the linear
theory. Our findings generalise known results in bimetric theory and allow us
to point out similarities and differences between the bi- and multimetric
models.
