Inverse eigenvalue problems appear in a wide variety of areas in the pure and applied mathematics. They have to do with the construction of a certain matrix from some spectral information. Associated with any inverse eigenvalue problem, there are two important issues: the existence of a solution and the construction of a solution matrix. The purpose of this chapter is to study the nonnegative inverse elementary divisors problem (hereafter, NIEDP) and its state of the art. The elementary divisors of a given matrix A are the characteristic polynomials of the Jordan blocks of the Jordan canonical form of A. The NIEDP looks for necessary and sufficient conditions for the existence of a nonnegative matrix with prescribed elementary divisors. Most of the content of this chapter is based on recent results published by the author and collaborators from the Mathematics Department at Universidad Catolica del Norte, Chile.