  A systematic study of (smooth, strong) cone structures $\C$ and
Lorentz-Finsler metrics $L$ is carried out. As a link between both notions,
cone triples $(\Omega,T, F)$, where $\Omega$ (resp. $T$) is a 1-form (resp.
vector field) with $\Omega(T)\equiv 1$ and $F$, a Finsler metric on $\ker
(\Omega)$, are introduced. Explicit descriptions of all the Finsler spacetimes
are given, paying special attention to stationary and static ones, as well as
to issues related to differentiability.
  In particular, cone structures $\C$ are bijectively associated with classes
of anisotropically conformal metrics $L$, and the notion of {\em cone geodesic}
is introduced consistently with both structures. As a non-relativistic
application, the {\em time-dependent} Zermelo navigation problem is posed
rigorously, and its general solution is provided.
