Bimetric and multimetric theories of gravity

It is well known that the interactions of the gravitational metric with other spin-2 fields are strongly constrained by the absence of ghost instabilities. In this talk I will outline the theoretical motivations for the study of such theories and then describe the ghost free theories of two interacting spin-2 fields (often called bimetric theories) and their known extensions to multiple spin-2 fields (multimetric theories). I will also describe an interesting geometrical aspect of such theories which is relevant for the existence of a consistent notion of space and time in the theory.