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.