In 1939 Weyl showed that the volume of spherical tubes around compact submanifolds M of Euclidean space depends solely on the induced Riemannian metric on M. We investigate whether this intrinsic nature of the tube volume can be preserved (a) for tubes with more general cross sections than the round ball, and (b) for tubes around submanifolds of Minkowski space. Joint work with Gert Heckman.