  We stratify families of projective and very affine hypersurfaces according to
their topological Euler characteristic. Our new algorithms compute all strata
using algebro-geometric techniques. For very affine hypersurfaces, we
investigate and exploit the relation to critical point computations. Euler
stratifications are relevant in particle physics and algebraic statistics. They
fully describe the dependence of the number of master integrals, respectively
the maximum likelihood degree, on kinematic or model parameters.
