  We describe the notion of a \emph{weighting} along a submanifold $N\subset
M$, and explore its differential-geometric implications. This includes a
detailed discussion of weighted normal bundles, weighted deformation spaces,
and weighted blow-ups. We give a description of weightings in terms of
subbundles of higher tangent bundles, which leads us to notions of
multiplicative weightings for Lie algebroids and Lie groupoids.
