Consider a small spherical tube around a compact submanifold M in Euclidean space. In 1939 Weyl showed that the volume of such a tube only depends on the radius of the tube and the intrinsic curvature of M. What happens for tubes with more complicated cross sections D? We will see that under sufficiently strong symmetry assumptions on D the tube volume turns out to be still intrinsic. This gives hope that causal tubes around spacelike submanifolds in Minkowski space also exhibit such nice properties. In the Lorentzian setting, however, the situation is more subtle. Joint work with Gert Heckman.