In this work, after a theoretical explanation of the monotone iteration method, there are presented several numerical experiments with this method, when applied to solve some nonlinear elliptic equations. It is shown that, in some cases, uniqueness of solution can also be verified through the numerical implementation of the method. It is also presented its application to cooperative elliptic systems. For all the examples Newton's method is also applied and a comparison between the monotone iteration method and Newton's method is made. (C) 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.3641700231248