  We consider Hamilton--Jacobi equations, where the Hamiltonian depends
discontinuously on both the spatial and temporal location. Our main results are
the existence and well--posedness of a viscosity solution to the Cauchy
problem. We define a viscosity solution by treating the discontinuities in the
coefficients analogously to ``internal boundaries''. By defining an appropriate
penalization function, we prove that viscosity solutions are unique. The
existence of viscosity solutions is established by showing that a sequence of
front tracking approximations is compact in $L^\infty$, and that the limits are
viscosity solutions.
