  We present a concise new definition of Finsler spacetimes that generalize
Lorentzian metric manifolds and provide consistent backgrounds for physics.
Extending standard mathematical constructions known from Finsler spaces we show
that geometric objects like the Cartan non-linear connection and its curvature
are well-defined almost everywhere on Finsler spacetimes, also on their null
structure. This allows us to describe the complete causal structure in terms of
timelike and null curves; these are essential to model physical observers and
the propagation of light. We prove that the timelike directions form an open
convex cone with null boundary as is the case in Lorentzian geometry. Moreover,
we develop action integrals for physical field theories on Finsler spacetimes,
and tools to deduce the corresponding equations of motion. These are applied to
construct a theory of electrodynamics that confirms the claimed propagation of
light along Finsler null geodesics.
