  We derive Frenet-type results and invariants of spatial curves immersed in
$3$-dimensional generalized Minkowski spaces, i.e., in linear spaces which
satisfy all axioms of finite dimensional real Banach spaces except for the
symmetry axiom. Further on, we characterize cylindrical helices and rectifying
curves in such spaces, and the computation of invariants is discussed, too.
Finally, we study how translations of unit spheres influence invariants of
spatial curves.
