  We define dilatations of general schemes and study their basic properties.
Dilatations of group schemes are -- in favorable cases -- again group schemes,
called N\'eron blowups. We give two applications to their cohomology in degree
zero (integral points) and degree one (torsors): we prove a canonical
Moy-Prasad isomorphism that identifies the graded pieces in the congruent
filtration of $G$ with the graded pieces in its Lie algebra $\mathfrak g$, and
we show that many level structures on moduli stacks of $G$-bundles are encoded
in torsors under N\'eron blowups of $G$.
