  The classical tools which ensure the completeness of vector fields and second
order differential equations for mechanical systems are revisited. Possible
extensions in three directions are discussed: infinite dimensional Banach and
Hilbert manifolds, Finsler metrics and pseudo-Riemannian spaces, including
links with some relativistic spacetimes. Special emphasis is taken in the
cleaning up of known techniques, the statement of open questions and the
exploration of prospective frameworks.
