Nonlinear stochastic difference equations appear in many branches of science and industry. Moreover, they arise naturally in the study of numerical solutions of stochastic differential equations. Our aim here is to study the stability of the numerical solutions of SDEs. More precisely, the issue which we address in this talk is: Can a numerical method reproduce the stability behavior of the underlying hybrid SDE? In this talk, we will first show that under the global Lipschitz condition plus some additional conditions the EM method can reproduce the almost sure exponential stability of the test hybrid SDEs. We will then point out that without this global Lipschitz condition the EM method may not preserve the almost sure exponential stability. We will further show that the backward EM method can capture almost sure exponential stability for a certain class of highly nonlinear hybrid SDEs.