The nonnegative inverse eigenvalue problem

Let σ=(λ1,…,λn) be a list of complex numbers. The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions in order that σ be the spectrum of an entrywise nonnegative n×n matrix. Our purpose is to give an overview of the NIEP, its history, modern results, and subproblems with particular emphasis on symmetric realizing matrices. A substantial bibliography is included.