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. ; Comment: Five new references included, plus some discussions and minor corrections; 28 pages, 2 figures. To appear in "Geometric Mechanics: The Legacy of Jerry Marsden", Fields Institute Communications Series