  We construct gravitational dynamics for Finsler spacetimes in terms of an
action integral on the unit tangent bundle. These spacetimes are
generalizations of Lorentzian metric manifolds which satisfy necessary
causality properties. A coupling procedure for matter fields to Finsler gravity
completes our new theory that consistently becomes equivalent to Einstein
gravity in the limit of metric geometry. We provide a precise geometric
definition of observers and their measurements, and show that the
transformations by means of which different observers communicate form a
groupoid that generalizes the usual Lorentz group. Moreover, we discuss the
implementation of Finsler spacetime symmetries. We use our results to analyze a
particular spacetime model that leads to Finsler geometric refinements of the
linearized Schwarzschild solution.
