  We present our Finsler spacetime formalism which extends the standard
formulation of Finsler geometry to be applicable in physics. Finsler spacetimes
are viable non-metric geometric backgrounds for physics; they guarantee well
defined causality, the propagation of light on a non-trivial null structure, a
clear notion of physical observers and the existence of physical field theories
determining the geometry of space-time dynamically in terms of an extended
gravitational field equation. Here we give the precise definition of Finsler
spacetimes and the notion of well-defined observers, their measurements, the
transformations between them and how to formulate action based field theories.
