The list Lambda = {lambda(1), lambda(2), ..., lambda(n)}, of complex numbers, is said to be realizable if it is the spectrum of an entrywise nonnegative matrix A. If A is diagonalizable, Lambda is said to be diagonalizably realizable. Lambda is universally realizable if it is realizable for each possible Jordan canonical form allowed by Lambda. Here, new realizability criteria are introduced. A diagonalizable version of a perturbation result by Rado is also proved. It allows to construct diagonalizable nonnegative matrices with prescribed spectrum. Criteria to decide the universal realizability of spectra are also established. (C) 2020 Elsevier Inc. All rights reserved.