In this thesis, we present an extended account of the real nonnegative inverse eigenvalue problem (RNIEP) based on an article by Carlos Marijuan, Miriam Pisonero and Ricardo L.Soto (A map of sufficient conditions for the real nonnegative inverse eigenvalue problem). This problem is originated from the inverse eigenvalue problem (IEP) which has a remarkable variety of applications in mathematical modeling. The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of complex numbers ? to be the spectrum of an entrywise nonnegative matrix. This problem has only been solved for n=3 by Loewy and London in 1978, and for the cases n=4 and n=5 for matrices of trace zero, by Reams in 1996, Laffey and Meehan in 1999. For the NIEP, in addition to the 3 basic necessary conditions, we have the JLL inequality due to Johnson, Loewy and London in 1978 and a refinement of it in the case of trace zero due to Laffey and Meehan in 1998. When ? is a list of real numbers, the problem changes to the RNIEP. This problem has only been solved for n?4 by Loewy and London . However, a number of sufficient conditions have been obtained which some of them have constructive proofs in the sense that one can explicitly construct nonnegative matrices realizing the prescribed real spectrum. This thesis has been arranged as follows: We begin by introducing the notation and basic concepts and some important applications of the IEP. We determine the necessary conditions under which a NIEP has a solution, in chapter 2. In chapter 3, we probe the NIEP in special solved cases which are quoted before.