  We employ the formalism of vanishing cycles and perverse sheaves to introduce
and study the vanishing cohomology of complex projective hypersurfaces. As a
consequence, we give upper bounds for the Betti numbers of projective
hypersurfaces, generalizing those obtained by different methods by Dimca in the
isolated singularities case, and by Siersma-Tib\u{a}r in the case of
hypersurfaces with a $1$-dimensional singular locus. We also prove a supplement
to the Lefschetz hyperplane theorem for hypersurfaces, which takes the
dimension of the singular locus into account, and we use it to give a new proof
of a result of Kato.
