  There are two well known tasks, related to Newton polyhedra: to study
invariants of singularities in terms of their Newton polyhedra, and to describe
Newton polyhedra of resultants and discriminants. We introduce so called
resultantal singularities, whose study in terms of Newton polyhedra unifies
these two tasks to a certain extent. As an application, we study topological
invariants of determinantal singularities and (co)vector fields on singular
varieties in terms of Newton polyhedra, and provide new formulations and proofs
for a number of well known results. The computations are based on certain
relative versions of the mixed volume and the
Kouchnirenko--Bernstein--Khovanskii formula. For the most part, this is a
significantly updated exposition of material from four recent papers by the
author.
