  We present a survey on Weil sums in which an additive character of a finite
field $F$ is applied to a binomial whose individual terms (monomials) become
permutations of $F$ when regarded as functions. Then we indicate how these Weil
sums are used in applications, especially how they characterize the
nonlinearity of power permutations and the correlation of linear recursive
sequences over finite fields. In these applications, one is interested in the
spectrum of Weil sum values that are obtained as the coefficients in the
binomial are varied. We review the basic properties of such spectra, and then
give a survey of current topics of research: Archimedean and non-Archimedean
bounds on the sums, the number of values in the spectrum, and the presence or
absence of zero in the spectrum. We indicate some important open problems and
discuss progress that has been made on them.
